#6915664

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907869
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	#30417	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1521 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)(17,18,23,26)(19,24,27,31)(20,28,30,32)(21,25,22,29);(1,3,7,4)(2,5,8,6)(9,10)(11,15)(12,14)(13,16)(17,19,23,27)(18,24,26,31)(20,29,30,25)(21,32,22,28);(1,4,7,3)(2,5,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,22,31,23,30,21,24)(18,19,25,28,26,27,29,32);(1,5,7,6)(2,3,8,4)(11,16)(13,15)(19,27)(20,30)(24,31)(28,32);(1,6,7,5)(2,3,8,4)(10,14)(13,15)(17,21,23,22)(18,25,26,29)(19,28,27,32)(20,24,30,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fg,d^2=f,e^2=fi,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=dfh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6822860);