#6407429

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle, prime radical	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=f,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,d^b=df,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6314625);