#6485914

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1374 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)(9,10)(11,15)(12,16)(13,14)(17,18,24,28)(19,31,36,45)(20,37,38,43)(21,27,32,42)(22,30,23,29)(25,35,44,34)(26,33,46,39)(40,48,41,47);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,19,23,35,24,36,22,34)(18,25,30,45,28,44,29,31)(20,21,39,41,38,32,33,40)(26,27,37,48,46,42,43,47);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,20,23,39,24,38,22,33)(18,26,30,37,28,46,29,43)(19,32,35,40,36,21,34,41)(25,42,45,47,44,27,31,48);(9,11,13,12)(10,15,14,16)(17,21,24,32)(18,27,28,42)(19,33,36,39)(20,34,38,35)(22,40,23,41)(25,43,44,37)(26,31,46,45)(29,47,30,48);(1,4,5,3)(2,6,8,7)(9,12,13,11)(10,15,14,16)(18,28)(20,38)(21,32)(25,44)(29,30)(31,45)(33,39)(40,41)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6393110);