#72336719

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 9846 conjugacy classes of subgroups; only the 3301 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:

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

Polycyclic presentation:

2^10|c^2=gj,e^2=hi,f^2=ghi,b^a=bg,c^a=cgj,c^b=cgh,d^c=dhi,e^a=egj,e^c=eghij,e^d=ehi,f^a=fj,f^c=fghi,f^d=fgi,f^e=fh

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336751
maximal normal subgroup of	#72336751

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.7*F.10);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.7*F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9*F.10);;
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.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);