#72330758

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	C₂²×#2678	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 6897 conjugacy classes of subgroups; only the 1023 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:

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

Polycyclic presentation:

2^10|b^2=fj,c^2=g,d^2=fgij,e^2=fghij,b^a=bf,c^a=cfi,c^b=cfj,d^a=dfgh,d^c=dfghj,e^a=eghi,e^b=eghi,e^c=efhi,e^d=ehij

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