#353213

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₄	C₂×C₄³
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1830 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)(9,10,14,18,15,19,13,28)(11,20,24,29,21,33,23,16)(12,17,27,32,25,31,26,38)(22,34,37,39,35,40,36,30);(2,4)(9,11)(10,16,19,29)(12,22)(13,23)(14,24)(15,21)(17,30,31,39)(18,20,28,33)(25,35)(26,36)(27,37)(32,34,38,40);(5,6,7,8)(11,21)(12,25)(16,29)(17,31)(20,33)(23,24)(26,27)(32,38);(9,12,14,27,15,25,13,26)(10,17,18,32,19,31,28,38)(11,22,24,37,21,35,23,36)(16,30,20,34,29,39,33,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=f,e^2=i,f^2=i,b^a=be,e^a=ei,g^a=gi,c^b=ci,d^c=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260409);