#6433748

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433042
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	#34633	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1968 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)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,22)(21,25)(23,26)(24,30)(28,32)(29,31);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,29,23,28,21,30)(18,24,25,32,26,31,20,27);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,21,26,23,25,22,18)(19,27,30,31,28,32,29,24);(10,14)(13,16)(19,28)(24,31)(27,32)(29,30);(17,21,23,22)(18,25,26,20)(19,29,28,30)(24,27,31,32)

Polycyclic presentation:

2^9|b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^c=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, 6340944);