#127037

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₄×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,e^2=h,f^2=g,h^2=i,b^a=bd,d^a=dg,e^b=eg

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34233);