#6456429

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fi,d^2=f,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363625);