#72334313

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

67193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

(9,10)(11,12)(13,14)(15,16)(17,18,20,19);(5,6)(7,8)(14,16)(18,19);(2,4)(6,8)(10,12)(13,14)(15,16)(17,19,20,18);(2,4)(6,8)(9,10,11,12)(13,14)(15,16);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)

Polycyclic presentation:

2^10|a^2=h,c^2=h,d^2=g,b^a=bj,c^a=cg,c^b=chi,d^a=dg,d^b=di,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334323, #72334326, C₂×#72334313, …
maximal normal subgroup of	#72334323, #72334326, C₂×#72334313, …
maximal quotient of	C₂×#72334313, #72335092

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