#6744024

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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1306 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)(4,8)(5,6)(9,10,12,13)(11,15,16,14)(17,18)(19,28)(20,33)(21,36)(22,25)(23,27)(24,31)(26,35)(29,38)(30,39)(32,42)(34,40)(37,41)(43,45)(44,48)(46,47);(10,13)(11,16)(17,19)(18,24)(20,30)(21,32)(22,31)(23,29)(25,28)(26,39)(27,42)(33,43)(34,44)(35,45)(36,38)(37,46)(40,47)(41,48);(2,6)(7,8)(9,11,12,16)(10,14,13,15)(18,25)(19,29)(21,23)(24,38)(26,33)(28,42)(30,44)(31,32)(34,37)(39,47)(43,48)(45,46);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20)(18,26)(19,30)(21,37)(22,35)(23,34)(24,39)(25,33)(27,41)(28,43)(29,44)(31,45)(32,46)(36,40)(38,47)(42,48);(1,4,5,3)(2,8,6,7)(17,21)(18,27)(19,31)(20,34)(22,23)(25,36)(26,40)(33,41)(35,37)(38,42)(39,43)(44,46)

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6651220);