#6374802

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1594 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,e^c=ei,e^d=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, 6281998);