#6468832

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1785 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,26,22,25,34,39)(19,27,29,36,30,40,33,23)(20,24,32,38,31,37,44,47)(28,41,43,46,42,48,45,35);(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,23,25,36)(20,28,31,42)(21,33,34,29)(24,35,37,46)(26,40,39,27)(32,45,44,43)(38,48,47,41);(2,5)(6,8)(10,14)(11,15)(17,20,22,31)(18,24,25,37)(19,28,30,42)(21,32,34,44)(23,35,36,46)(26,38,39,47)(27,41,40,48)(29,43,33,45);(11,15)(13,16)(20,31)(24,37)(28,42)(32,44)(35,46)(38,47)(41,48)(43,45);(1,3,4,7)(2,6,5,8)(18,25)(23,36)(24,37)(26,39)(27,40)(35,46)(38,47)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376028);