#72328802

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
prime radical	C₂×#14716	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 6881 conjugacy classes of subgroups; only the 1003 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,22,23,20)(25,26,29,30)(27,31,32,28);(1,2,3,4)(6,8)(10,11)(12,16)(19,23)(20,22)(25,27)(26,28)(29,32)(30,31);(19,23)(20,22)(27,32)(28,31);(1,2,3,4)(5,6,7,8)(9,11,13,10)(12,14,16,15)(17,19,21,23)(18,22,24,20)(25,27)(26,28)(29,32)(30,31);(1,2)(3,4)(9,12,13,16)(10,14,11,15)(17,20,21,22)(18,23,24,19)(25,28)(26,27)(29,31)(30,32)

Polycyclic presentation:

2^10|a^2=j,b^2=i,d^2=hij,e^2=gj,b^a=bg,c^b=cf,d^a=dh,d^b=dhj,d^c=dj,e^a=efj,e^b=egij,e^c=ej,e^d=efgi

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

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