#6519370

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	#2433	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1587 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)(3,6)(4,5)(7,8)(18,24)(19,26)(21,22)(25,29)(27,28)(31,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,19,23,26,21,24)(20,28,32,29,30,25,31,27);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,19,21,18,23,24,22,26)(20,29,31,28,30,27,32,25);(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,27)(21,31)(22,32)(23,30)(24,29)(26,28);(10,13)(14,16)(20,30)(25,28)(27,29)(31,32)

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,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, 6426566);