#8994503

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

4237 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=i,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8901699);