#6461980

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1492 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)(4,7)(5,6)(17,18,23,35,22,25,21,27)(19,28,34,24,31,41,30,37)(20,26,36,46,32,39,33,40)(29,42,45,38,43,48,44,47);(2,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,31)(18,24,25,37)(20,29,32,43)(21,34,23,30)(26,38,39,47)(27,41,35,28)(33,45,36,44)(40,48,46,42);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(18,25)(24,37)(26,39)(27,35)(28,41)(38,47)(40,46)(42,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,20,22,32)(18,26,25,39)(19,29,31,43)(21,33,23,36)(24,38,37,47)(27,40,35,46)(28,42,41,48)(30,44,34,45);(18,25)(20,32)(24,37)(27,35)(28,41)(29,43)(33,36)(44,45)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=ghi,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6369176);