#8633841

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

4178 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8541037);