#72334404

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁵
commutator	C₂⁴	C₂⁴×C₄
prime radical	C₂⁷	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

70917 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fgi,c^2=j,b^a=bfg,c^a=ch,c^b=ch,d^a=dgij,d^b=dgij,e^a=efghj,e^b=efghj

Conjugacy classes & character table

304 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334414, #72334417, #72334429, …
maximal normal subgroup of	#72334414, #72334417, #72334429, …
maximal quotient of	C₂×#72334404

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