#6506621

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37418	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1718 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6413817);