#8118849

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

4056 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026045);