#6651309

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1449 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=h,e^2=fi,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,e^c=ei,g^c=gi,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, 6558505);