#358043

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1786 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)(9,10,14,18,15,19,13,29)(11,20,23,31,24,34,22,16)(12,25,28,32,26,39,27,17)(21,35,38,40,36,30,37,33);(5,6)(7,8)(9,11,13,22,15,24,14,23)(10,16,18,20,19,31,29,34)(12,21,27,37,26,36,28,38)(17,30,25,33,32,35,39,40);(2,4)(6,8)(9,12,13,27,15,26,14,28)(10,17,18,25,19,32,29,39)(11,21,22,37,24,36,23,38)(16,30,20,33,31,35,34,40);(12,26)(17,32)(21,36)(25,39)(27,28)(30,35)(33,40)(37,38)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265239);