#8093086

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

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

6192 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263064, #72308577, #72308654, …
maximal normal subgroup of	#72263064, #72308577, #72308654, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000282);