#6761939

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682380
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	#37656	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1886 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)(11,13)(12,14)(15,16)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,34)(28,42)(29,38)(30,39)(31,40)(33,41)(35,46)(37,43)(44,48)(45,47);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,31)(18,24,27,40)(20,28,35,45)(21,30,22,29)(25,37,41,48)(26,39,36,38)(32,42,46,47)(33,44,34,43);(10,14)(11,15)(17,20,21,33,23,35,22,34)(18,25,36,46,27,41,26,32)(19,28,30,44,31,45,29,43)(24,37,38,47,40,48,39,42);(2,6)(5,7)(17,21,23,22)(18,26,27,36)(19,29,31,30)(20,33,35,34)(24,38,40,39)(25,32,41,46)(28,43,45,44)(37,47,48,42);(3,8)(5,7)(11,15)(13,16)(17,21,23,22)(18,26,27,36)(19,30,31,29)(20,33,35,34)(24,39,40,38)(25,32,41,46)(28,44,45,43)(37,42,48,47)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6669135);