#354671

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,4)(5,6,8,7)(9,10)(11,16)(12,24)(13,28)(14,18)(15,19)(17,26)(20,33)(21,37)(22,30)(23,31)(25,38)(27,32)(29,35)(34,40)(36,39);(2,4)(9,11,14,22,15,23,13,21)(10,16,18,30,19,31,28,37)(12,20,27,36,25,34,26,35)(17,29,24,33,32,39,38,40);(9,12,13,26,15,25,14,27)(10,17,18,24,19,32,28,38)(11,20,21,35,23,34,22,36)(16,29,30,33,31,39,37,40);(5,7,8,6)(12,25)(17,32)(20,34)(24,38)(26,27)(29,39)(33,40)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,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, 261867);