#6508414

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
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	#25736	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh

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, 6415610);