#6691135

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 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)(3,6)(4,8)(5,7)(9,10)(11,13)(12,15)(14,16)(17,18,23,28)(19,24,32,40)(20,25,22,27)(21,26,35,43)(29,37,30,38)(31,39,46,48)(33,41,36,44)(34,42,45,47);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,24)(20,30)(21,34)(22,29)(23,32)(25,38)(26,42)(27,37)(28,40)(31,33)(35,45)(36,46)(39,41)(43,47)(44,48);(1,4,5,3)(2,6,7,8)(10,11)(14,15)(19,29)(20,22)(24,37)(25,27)(30,32)(31,34)(33,36)(38,40)(39,42)(41,44)(45,46)(47,48);(9,12,13,16)(10,14,11,15)(17,20,23,22)(18,25,28,27)(19,30,32,29)(21,33,35,36)(24,38,40,37)(26,41,43,44)(31,45,46,34)(39,47,48,42);(17,21,23,35)(18,26,28,43)(19,31,32,46)(20,33,22,36)(24,39,40,48)(25,41,27,44)(29,34,30,45)(37,42,38,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6598331);