#6471790

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	#23857	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1845 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)(11,13)(12,14)(15,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,30)(28,31)(29,32);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,21)(18,23,26,24)(20,28,29,27)(25,31,32,30);(11,15)(13,16)(18,24)(19,21)(23,26)(25,31)(27,28)(30,32);(11,15)(13,16)(17,20,22,29)(18,25,26,32)(19,27,21,28)(23,30,24,31);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(18,26)(20,29)(23,24)(27,28)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,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, 6378986);