#378898

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,8)(5,6)(9,10,15,20,16,22,14,21)(11,17,13,19,25,33,24,32)(12,26,28,18,29,35,30,36)(23,37,27,31,38,34,39,40);(2,6)(7,8)(9,11,14,24,16,25,15,13)(10,17,21,32,22,33,20,19)(12,27,28,38,29,39,30,23)(18,34,35,40,36,37,26,31);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,23)(13,27)(14,30)(15,28)(16,29)(17,31)(19,34)(20,35)(21,26)(22,36)(24,39)(25,38)(32,37)(33,40);(1,4,5,3)(2,7,6,8)(9,13,15,25,16,24,14,11)(10,19,20,33,22,32,21,17)(12,27,28,38,29,39,30,23)(18,34,35,40,36,37,26,31)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=e,d^2=eh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^c=fi,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, 286094);