#6764570

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
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	#29738	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1541 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)(11,13)(12,14)(15,16)(17,18,22,28)(19,29,25,24)(20,31,32,30)(21,27,23,26);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,27)(21,31)(22,25)(23,30)(26,32)(28,29);(10,14)(11,15)(17,20,23,29,22,32,21,24)(18,25,27,31,28,19,26,30);(2,5)(6,8)(17,21,22,23)(18,26,28,27)(19,30,25,31)(20,29,32,24);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,21,22,23)(18,27,28,26)(19,31,25,30)(20,29,32,24)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6671766);