#7623173

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1688 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)(11,18)(12,17)(13,19)(14,21)(15,20)(16,23)(22,30)(24,36)(25,37)(26,33)(27,32)(28,34)(29,38)(31,39)(35,40);(2,5)(6,8)(9,11,15,26,13,25,29,27,16,24,31,40,30,28,14,12)(10,17,21,34,22,35,39,36,23,32,38,37,19,33,20,18);(1,3,4,7)(2,6,5,8)(9,12,14,28,30,40,31,24,16,27,29,25,13,26,15,11)(10,18,20,33,19,37,38,32,23,36,39,35,22,34,21,17);(11,24)(12,27)(17,32)(18,36)(25,28)(26,40)(33,35)(34,37);(9,13,16,30)(10,19,23,22)(11,25,24,28)(12,26,27,40)(14,15,29,31)(17,33,32,35)(18,37,36,34)(20,38,39,21)

Polycyclic presentation:

2^9|b^2=g,c^2=f,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,h^a=hi,d^b=di,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, 7530369);