#6465336

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1376 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)(3,7)(4,6)(5,8)(17,18,24,30)(19,31,36,44)(20,37,38,43)(21,27,32,42)(22,29,23,28)(25,35,45,34)(26,33,46,39)(40,48,41,47);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,19,23,35,24,36,22,34)(18,25,29,44,30,45,28,31)(20,21,39,41,38,32,33,40)(26,27,37,48,46,42,43,47);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,22,33,24,38,23,39)(18,26,28,43,30,46,29,37)(19,32,34,41,36,21,35,40)(25,42,31,48,45,27,44,47);(9,11,12,16)(10,14,13,15)(17,21,24,32)(18,27,30,42)(19,33,36,39)(20,34,38,35)(22,40,23,41)(25,43,45,37)(26,31,46,44)(28,47,29,48);(1,4,5,3)(2,6,8,7)(20,38)(21,32)(26,46)(27,42)(33,39)(37,43)(40,41)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=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, 6372532);