#6506290

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1401 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)(3,8)(4,6)(5,7)(17,18,24,28,23,30,22,29)(19,25,38,41,34,43,33,42)(20,27,35,26,21,37,36,44)(31,40,45,39,32,47,46,48);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,34)(18,25,30,43)(20,31,21,32)(22,33,24,38)(26,39,44,48)(27,40,37,47)(28,41,29,42)(35,45,36,46);(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,26,30,44)(19,31,34,32)(22,35,24,36)(25,39,43,48)(27,29,37,28)(33,45,38,46)(40,42,47,41);(11,16)(13,15)(17,21,23,20)(18,27,30,37)(19,32,34,31)(22,35,24,36)(25,40,43,47)(26,29,44,28)(33,45,38,46)(39,42,48,41);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,23,24)(18,28,30,29)(19,33,34,38)(20,35,21,36)(25,41,43,42)(26,27,44,37)(31,45,32,46)(39,40,48,47)

Polycyclic presentation:

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

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, 6413486);