#8746585

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

4145 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,23)(20,25)(21,26)(22,27)(24,28)(29,31)(30,32)(33,34,38,37)(35,40,43,39)(36,44,45,41)(42,46,48,47);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,31)(26,32)(33,35,38,43)(34,39,37,40)(36,42,45,48)(41,46,44,47);(2,3)(7,9)(8,10)(15,16)(19,28)(23,24)(29,32)(30,31)(33,35,38,43)(34,39,37,40)(36,42,45,48)(41,46,44,47);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,19)(18,24)(20,29)(21,31)(22,28)(23,27)(25,30)(26,32)(33,35)(34,40)(36,42)(37,39)(38,43)(41,47)(44,46)(45,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20)(18,25)(19,29)(21,27)(22,26)(23,31)(24,30)(28,32)(33,36,38,45)(34,41,37,44)(35,42,43,48)(39,46,40,47);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,25)(18,26,27,20)(19,30,28,31)(23,32,24,29)(33,37,38,34)(35,40,43,39)(36,41,45,44)(42,46,48,47)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653781);