#6700512

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6607708);