#6374088

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator, socle	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1612 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)(11,16)(12,14)(13,15)(17,18,21,25,22,20,24,29)(19,23,27,30,26,28,31,32);(2,5)(6,8)(17,19)(18,23)(20,28)(21,27)(22,26)(24,31)(25,30)(29,32);(9,11,13,12)(10,14,15,16)(19,26)(23,28)(27,31)(30,32);(9,12,13,11)(10,14,15,16)(17,20,21,29,22,18,24,25)(19,23,27,30,26,28,31,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=gh,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ci,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281284);