#6466746

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373942);