#6506610

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1657 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,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, 6413806);