#7622731

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂⁴
commutator, socle	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂²×#41	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1427 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,e^2=f,g^2=h,h^2=i,b^a=bf,d^a=df,c^b=cfi

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529927);