#8618090

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

4156 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,e^2=i,f^2=g,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, 8525286);