#7007812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1353 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6915008);