#8755033

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

4155 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)(3,9,4,7)(5,13,14,10)(11,15,12,16)(17,18)(19,24)(20,22)(21,30)(23,25)(26,29)(27,32)(28,31)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,3,6,4)(2,7,8,9)(5,12,14,11)(10,16,13,15)(17,19)(18,24)(20,29)(21,27)(22,26)(23,28)(25,31)(30,32)(33,35,37,43)(34,38,40,41)(36,44,45,42)(39,47,48,46);(2,8)(7,9)(10,13)(15,16)(18,25)(19,28)(20,27)(26,30)(33,34,37,40)(35,38,43,41)(36,39,45,48)(42,46,44,47);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,16)(13,14)(17,20,23,27)(18,22,25,32)(19,29,28,21)(24,26,31,30)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48);(1,4,6,3)(2,9,8,7)(5,12,14,11)(10,16,13,15)(17,21,23,29)(18,26,25,30)(19,27,28,20)(22,31,32,24)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,5,6,14)(2,10,8,13)(3,11,4,12)(7,15,9,16)(17,22)(18,27)(19,30)(20,25)(21,31)(23,32)(24,29)(26,28)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662229);