#6490786

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469983
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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1538 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)(9,10)(11,14)(12,15)(13,16)(17,18,23,37,22,26,21,27)(19,28,36,24,30,43,31,40)(20,32,38,42,34,25,35,47)(29,45,48,41,33,44,46,39);(2,6)(7,8)(9,11,14,10)(12,15,13,16)(17,19,22,30)(18,24,26,40)(20,33,34,29)(21,36,23,31)(25,41,32,39)(27,43,37,28)(35,46,38,48)(42,44,47,45);(1,3,5,4)(2,7,6,8)(10,11)(15,16)(17,20,22,34)(18,25,26,32)(19,29,30,33)(21,35,23,38)(24,39,40,41)(27,42,37,47)(28,44,43,45)(31,46,36,48);(9,12,14,13)(10,15,11,16)(19,30)(24,40)(28,43)(29,33)(31,36)(39,41)(44,45)(46,48);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,15,11,16)(18,26)(20,34)(24,40)(27,37)(28,43)(29,33)(35,38)(46,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=ghi,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6397982);