#6468316

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1509 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)(9,10,13,16)(11,14,12,15)(17,18)(19,26)(20,23)(21,25)(22,31)(24,27)(28,30)(29,32);(11,12)(14,15)(17,19,21,28,23,29,22,27)(18,24,31,32,20,30,25,26);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,20,23,18)(19,24,29,30)(21,25,22,31)(26,28,32,27);(9,12,13,11)(10,15,16,14)(17,21,23,22)(18,25,20,31)(19,27,29,28)(24,32,30,26);(1,3,4,7)(2,6,5,8)(17,21,23,22)(18,25,20,31)(19,28,29,27)(24,26,30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375512);