#7020052

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#45245	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1901 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)(19,23)(20,31)(21,25)(22,27)(24,28)(26,30)(29,39)(32,41)(33,47)(34,38)(35,48)(36,43)(37,44)(40,45)(42,46);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,27,26)(20,32,34,28)(21,29,37,36)(24,31,41,38)(25,39,44,43)(33,46,35,45)(40,47,42,48);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,30,32,22,34,19,28)(18,24,23,38,27,41,26,31)(21,35,36,45,37,33,29,46)(25,42,39,47,44,40,43,48);(3,8)(6,7)(11,16)(13,15)(17,21)(18,25)(19,29)(20,33)(22,37)(23,39)(24,40)(26,43)(27,44)(28,45)(30,36)(31,47)(32,46)(34,35)(38,48)(41,42);(10,14)(13,15)(17,19,22,30)(18,26,27,23)(20,32,34,28)(21,36,37,29)(24,38,41,31)(25,39,44,43)(33,45,35,46)(40,47,42,48)

Polycyclic presentation:

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

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, 6927248);