#6650282

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636940
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1190 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(33,34,37,40)(35,42,45,48)(36,39,47,41)(38,44,43,46);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,21,25)(18,22,24,29)(20,27,28,26)(23,31,32,30)(33,34)(35,43)(36,41)(37,40)(38,45)(39,47)(42,44)(46,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(19,25)(22,29)(26,27)(30,31)(33,35,36,46,37,45,47,44)(34,38,41,42,40,43,39,48);(2,9)(3,12)(8,14)(11,13)(17,20,21,28)(18,23,24,32)(19,26,25,27)(22,30,29,31)(33,34)(35,43)(36,41)(37,40)(38,45)(39,47)(42,44)(46,48);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,24)(22,29)(23,32)(30,31)(33,36,37,47)(34,39,40,41)(35,44,45,46)(38,42,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=fgi,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6557478);