#6741916

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	#30414	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1572 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,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,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, 6649112);