#8718245

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

4169 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,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, 8625441);