#6508607

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	#28319	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1643 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,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=ehi,d^c=dh,e^d=ei

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, 6415803);