#6488942

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
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

1345 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)(11,14)(12,15)(13,16)(17,18,22,25)(19,27,29,43)(20,31,33,24)(21,35,36,26)(23,37,39,30)(28,44,32,42)(34,41,45,47)(38,46,40,48);(11,12)(14,15)(17,19,21,30,22,29,36,37)(18,23,26,27,25,39,35,43)(20,32,34,48,33,28,45,46)(24,40,41,44,31,38,47,42);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,20,22,33)(18,24,25,31)(19,28,29,32)(21,34,36,45)(23,38,39,40)(26,41,35,47)(27,42,43,44)(30,46,37,48);(9,12,13,11)(10,15,16,14)(19,29)(23,39)(27,43)(28,32)(30,37)(38,40)(42,44)(46,48);(1,3,4,7)(2,6,5,8)(10,16)(14,15)(18,25)(20,33)(23,39)(26,35)(27,43)(28,32)(34,45)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=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, 6396138);