#353141

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂⁵×C₄²

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	C₂²×C₄⋊C₄
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2402 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)(11,21,23,16,25,30,24,31)(12,17,27,34,26,32,28,33)(22,35,37,29,36,40,38,39);(2,4)(9,11,14,24,15,25,13,23)(10,16,19,30,20,31,18,21)(12,22,28,38,26,36,27,37)(17,29,34,40,32,39,33,35);(5,6,7,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,26)(16,29)(18,33)(19,34)(20,32)(21,35)(23,37)(24,38)(25,36)(30,40)(31,39);(12,26)(17,32)(22,36)(27,28)(29,39)(33,34)(35,40)(37,38)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=i,f^2=i,g^2=i,b^a=be,e^a=ei,g^a=gi,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, 260337);