#8139499

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

4045 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:

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

Polycyclic presentation:

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

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, 8046695);