#357173

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1152 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)(9,10,14,20,15,21,29,19)(11,22,13,18,25,36,24,16)(12,17,27,33,28,34,39,32)(23,37,30,35,38,40,26,31);(5,6,8,7)(9,11,15,25)(10,16,21,18)(12,23,28,38)(13,29,24,14)(17,31,34,35)(19,36,20,22)(26,27,30,39)(32,40,33,37);(2,4)(5,7,8,6)(9,12,14,27,15,28,29,39)(10,17,20,33,21,34,19,32)(11,23,13,30,25,38,24,26)(16,31,22,37,18,35,36,40);(9,13,29,11,15,24,14,25)(10,18,19,22,21,16,20,36)(12,26,39,38,28,30,27,23)(17,31,32,40,34,35,33,37)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gh,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,d^b=de,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264369);