#8718227

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
prime radical	#52690	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4013 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)(17,18,23,27)(19,24,28,31)(20,25,29,30)(21,22,26,32)(33,34)(35,40)(36,45)(37,42)(38,39)(41,46)(43,44)(47,48);(2,3)(7,9)(8,10)(15,16)(17,19,23,28)(18,24,27,31)(20,21,29,26)(22,30,32,25)(33,34)(35,43)(36,41)(37,42)(38,39)(40,44)(45,46)(47,48);(2,3)(7,9)(8,10)(15,16)(19,28)(20,29)(24,31)(25,30)(34,39)(35,44)(41,45)(42,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,23,29)(18,25,27,30)(19,21,28,26)(22,31,32,24)(33,35)(34,40)(36,42)(37,45)(38,44)(39,43)(41,47)(46,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21)(18,22)(19,29)(20,28)(23,26)(24,30)(25,31)(27,32)(33,36)(34,41)(35,42)(37,43)(38,46)(39,45)(40,47)(44,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22)(18,26)(19,30)(20,24)(21,27)(23,32)(25,28)(29,31)(33,37,38,47)(34,42,39,48)(35,41,44,45)(36,40,46,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8625423);