#8650982

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

4190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8558178);