#6844274

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	#42746	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fhi,d^2=gi,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6751470);