#8115484

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8022680);