#354670

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
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×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1094 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)(13,29)(14,18)(15,19)(17,26)(20,35)(21,31)(22,38)(23,32)(25,39)(27,34)(28,40)(30,36)(33,37);(2,4)(9,11,14,23,15,21,13,22)(10,16,18,32,19,31,29,38)(12,25,27,36,28,20,26,37)(17,33,24,39,34,30,40,35);(9,12,13,26,15,28,14,27)(10,17,18,24,19,34,29,40)(11,20,22,36,21,25,23,37)(16,30,32,35,31,33,38,39);(5,7,8,6)(11,21)(16,31)(20,25)(22,23)(30,33)(32,38)(35,39)(36,37)

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,c^b=ci,d^b=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, 261866);