#72334409

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

60705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^10|a^2=j,c^2=fgh,d^2=fhij,b^a=bf,c^a=cg,c^b=ch,d^a=dg,d^b=dh,d^c=dgi,e^c=efgij,e^d=efgij

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334418, #72334429, C₂×#72334409, …
maximal normal subgroup of	#72334418, #72334429, C₂×#72334409, …
maximal quotient of	C₂×#72334409, #72337397

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