#72232629

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

20077 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^10|c^2=hi,b^a=bg,c^a=cj,c^b=ch,d^a=dj,e^a=ehij,f^a=fg,f^d=fij,f^e=fg

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232676, #72233148
maximal normal subgroup of	#72232676, #72233148

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