#8230705

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

4060 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8137901);