#181109

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,f^2=i,g^2=hi,b^a=be,e^a=eh,f^b=fh,d^c=di

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88305);