#6434848

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432838
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1452 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)(3,8)(4,6)(5,7)(10,14)(11,12)(17,18,23,19)(20,28,30,32)(21,26,22,25)(24,31,27,29);(1,3,5,4)(2,6,7,8)(9,10)(11,16)(12,15)(13,14)(17,19,23,18)(20,24,30,27)(21,26,22,25)(28,31,32,29);(9,11,13,12)(10,15,14,16)(17,20,23,30)(18,24,19,27)(21,29,22,31)(25,28,26,32);(1,4,5,3)(2,6,7,8)(9,12,13,11)(10,15,14,16)(18,19)(24,27)(25,26)(28,32);(17,21,23,22)(18,25,19,26)(20,29,30,31)(24,28,27,32)

Polycyclic presentation:

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

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, 6342044);