#6447452

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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	#24250	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1805 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)(17,18,21,25)(19,27,29,23)(20,24,30,31)(22,26,28,32);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(19,20)(22,28)(23,31)(24,27)(26,32)(29,30);(9,12,13,11)(10,14,15,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,26)(25,27)(31,32);(17,20,21,30)(18,24,25,31)(19,28,29,22)(23,26,27,32);(1,3,4,7)(2,6,5,8)(18,25)(23,27)(24,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354648);