#6695953

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1549 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)(11,14)(12,13)(15,16)(17,18)(19,24)(20,30)(21,23)(22,26)(25,32)(27,40)(28,44)(29,36)(31,41)(33,42)(34,37)(35,38)(39,45)(43,47)(46,48);(1,3,4,8)(2,5,6,7)(18,23)(19,27)(20,31)(22,34)(24,38)(25,41)(28,45)(29,35)(30,42)(32,33)(36,40)(47,48);(10,13)(14,16)(17,19,21,29)(18,24,23,36)(20,28,32,45)(22,35,34,27)(25,39,30,44)(26,38,37,40)(31,46,33,43)(41,48,42,47);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,23)(20,32)(24,36)(26,37)(28,45)(31,33)(38,40)(43,46);(3,8)(5,7)(17,20,21,32)(18,25,23,30)(19,28,29,45)(22,31,34,33)(24,39,36,44)(26,42,37,41)(27,43,35,46)(38,47,40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6603149);