#72336910

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 9827 conjugacy classes of subgroups; only the 3321 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(6,10)(11,12)(14,15)(16,20)(22,25)(26,27);(5,6)(7,11)(8,12)(9,10)(21,22)(23,26)(24,25)(27,28);(2,3)(7,8)(11,12)(13,14,17,15)(16,19,20,18)(21,22,24,25)(23,26,28,27);(2,3)(5,7,9,8)(6,11,10,12)(13,14,17,15)(16,19,20,18)(21,22)(23,27)(24,25)(26,28);(1,2,4,3)(5,8,9,7)(6,12,10,11)(13,15,17,14)(16,19,20,18)(21,22)(23,27)(24,25)(26,28);(1,3,4,2)(5,6)(7,12)(8,11)(9,10)(13,16,17,20)(14,18,15,19)(21,23)(22,26)(24,28)(25,27)

Polycyclic presentation:

2^10|c^2=hj,d^2=j,e^2=hij,f^2=gij,b^a=bh,c^a=chj,c^b=cg,d^a=dhj,d^c=dh,e^a=ehj,e^c=ei,e^d=ehi,f^a=fj,f^c=fgij,f^d=fghij,f^e=fh

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336936
maximal normal subgroup of	#72336936

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.8*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);