#8103484

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5458 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280721, #72280763
maximal normal subgroup of	#72280721, #72280763
maximal quotient of	#72281188

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8010680);