#6467046

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	#28475	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1640 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)(9,10,13,15)(11,14,12,16)(17,18,21,23)(19,25,28,20)(22,24,27,30)(26,31,32,29);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(19,26)(20,29)(22,27)(24,30)(25,31)(28,32);(9,12,13,11)(10,14,15,16)(17,19,27,32,21,28,22,26)(18,20,30,31,23,25,24,29);(17,20,22,29,21,25,27,31)(18,19,24,26,23,28,30,32);(1,3,4,7)(2,6,5,8)(18,23)(20,25)(24,30)(29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6374242);