#6593373

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6500569);