#8228663

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

4261 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,24)(20,26)(21,29)(22,28)(23,25)(27,32)(30,31);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,19,23,22)(18,24,25,28)(20,27,31,29)(21,26,32,30)(33,34,38,39)(35,37,41,46)(36,43,45,40)(42,48,47,44);(4,12)(7,14)(10,16)(13,15)(18,25)(19,22)(26,30)(27,29)(34,39)(37,46)(40,43)(44,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20,23,31)(18,26,25,30)(19,29,22,27)(21,28,32,24)(33,35,38,41)(34,37,39,46)(36,42,45,47)(40,44,43,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,21,23,32)(18,27,25,29)(19,30,22,26)(20,28,31,24)(33,36,38,45)(34,40,39,43)(35,42,41,47)(37,44,46,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,22,23,19)(18,28,25,24)(20,27,31,29)(21,26,32,30)(33,37)(34,41)(35,39)(36,44)(38,46)(40,47)(42,43)(45,48)

Polycyclic presentation:

2^9|b^2=gi,d^2=i,e^2=i,f^2=h,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, 8135859);