#354931

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×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	C₂²×C₄⋊C₄
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1546 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)(9,10,13,19,15,20,14,18)(11,21,23,16,25,31,24,32)(12,17,28,36,27,34,29,35)(22,37,38,30,26,40,39,33);(5,6,8,7)(9,11,14,24,15,25,13,23)(10,16,19,31,20,32,18,21)(12,26,29,38,27,22,28,39)(17,33,36,37,34,30,35,40);(2,4)(9,12)(10,17)(11,22)(13,28)(14,29)(15,27)(16,30)(18,35)(19,36)(20,34)(21,37)(23,38)(24,39)(25,26)(31,40)(32,33);(5,7,8,6)(12,27)(17,34)(22,26)(28,29)(30,33)(35,36)(37,40)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=ci,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, 262127);