#6467975

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,13)(15,16)(17,18,21,25)(19,27,30,23)(20,24,33,38)(22,26,32,37)(28,39,42,34)(29,40,43,35)(31,36,44,46)(41,47,48,45);(1,3,4,8)(2,5,6,7)(19,28)(20,31)(22,32)(23,34)(24,36)(26,37)(27,39)(29,43)(30,42)(33,44)(35,40)(38,46);(10,13)(14,16)(17,19,22,28,21,30,32,42)(18,23,26,34,25,27,37,39)(20,29,31,41,33,43,44,48)(24,35,36,45,38,40,46,47);(17,20,32,44,21,33,22,31)(18,24,37,46,25,38,26,36)(19,29,42,48,30,43,28,41)(23,35,39,47,27,40,34,45);(9,11,12,15)(10,14,13,16)(19,30)(23,27)(28,42)(29,43)(34,39)(35,40)(41,48)(45,47)

Polycyclic presentation:

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

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, 6375171);