#354742

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fi,d^2=fg,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261938);