#365110

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂×C₄×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1970 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)(5,6,7,8)(9,10,15,20,13,18,14,19)(11,16,23,32,22,31,28,35)(12,24,27,36,26,17,29,34)(21,37,39,33,38,30,25,40);(2,4)(9,11,14,28,13,22,15,23)(10,16,19,35,18,31,20,32)(12,25,29,38,26,39,27,21)(17,33,36,37,24,40,34,30);(6,8)(9,12)(10,17)(11,21)(13,26)(14,29)(15,27)(16,30)(18,24)(19,36)(20,34)(22,38)(23,39)(25,28)(31,37)(32,40)(33,35);(2,4)(12,26)(17,24)(21,38)(25,39)(27,29)(30,37)(33,40)(34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272306);