#7005578

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1500 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)(17,18,22,27,23,28,37,43)(19,29,32,46,33,24,38,41)(20,26,35,42,21,36,34,25)(30,45,48,39,31,40,47,44);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,33)(18,24,28,29)(20,30,21,31)(22,32,37,38)(25,39,42,44)(26,40,36,45)(27,41,43,46)(34,47,35,48);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,25,28,42)(19,30,33,31)(22,34,37,35)(24,39,29,44)(26,27,36,43)(32,47,38,48)(40,41,45,46);(3,8)(6,7)(11,16)(13,15)(17,21,23,20)(18,26,28,36)(19,31,33,30)(22,34,37,35)(24,40,29,45)(25,27,42,43)(32,47,38,48)(39,41,44,46);(10,14)(13,15)(17,22,23,37)(18,27,28,43)(19,32,33,38)(20,34,21,35)(24,41,29,46)(25,36,42,26)(30,47,31,48)(39,45,44,40)

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,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, 6912774);