#72329347

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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 10330 conjugacy classes of subgroups; only the 3333 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:

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

Polycyclic presentation:

2^10|c^2=h,e^2=j,f^2=hij,b^a=bi,c^a=cij,c^b=cj,d^b=dh,d^c=dh,e^a=eij,e^b=ej,e^c=ehj,e^d=eh,f^d=fhij,f^e=fg

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329448
maximal normal subgroup of	#72329448
maximal quotient of	#72331244

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