#6489456

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1323 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:

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

Polycyclic presentation:

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

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, 6396652);