#6457414

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1666 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)(9,10,12,14)(11,13,15,16)(17,18,23,27)(19,24,25,31)(20,30,32,29)(21,26,22,28);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,28)(21,30)(22,29)(23,25)(26,32)(27,31);(11,15)(13,16)(17,20,22,24,23,32,21,31)(18,25,26,30,27,19,28,29);(1,3,4,7)(2,6,5,8)(11,15)(13,16);(17,21,23,22)(18,26,27,28)(19,29,25,30)(20,31,32,24)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364610);