#72329283

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 10273 conjugacy classes of subgroups; only the 3421 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)(13,14)(15,20)(16,18)(17,19);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,17)(19,20)(21,22)(23,27)(24,26)(25,28);(21,23)(22,25)(24,28)(26,27);(7,11)(10,12)(15,16)(18,20)(22,26)(25,27);(2,4)(5,6)(7,12)(8,9)(10,11)(13,16,17,15)(14,18,19,20)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,14)(15,20)(16,18)(17,19)(21,23)(22,25)(24,28)(26,27)

Polycyclic presentation:

2^10|e^2=j,f^2=ij,b^a=bi,c^b=ch,d^b=dhj,e^a=eij,e^b=ej,e^d=ehj,f^a=fij,f^b=fhj,f^d=fij,f^e=fg

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329448
maximal normal subgroup of	#72329448
maximal quotient of	#72331147

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