#8630796

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

4022 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)(3,8)(4,9)(5,7)(6,10)(11,15)(12,16)(13,14)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(34,39)(35,42)(36,44)(46,47);(1,3,7,11)(2,8,5,15)(4,12,14,10)(6,9,16,13)(17,19,22,27)(18,23,21,25)(20,26,29,32)(24,30,28,31)(33,34)(35,37)(36,45)(38,39)(40,44)(41,42)(43,46)(47,48);(2,5)(3,11)(9,13)(10,12)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,35,38,42)(34,37,39,41)(36,43,44,48)(40,47,45,46);(1,4,7,14)(2,9,5,13)(3,10,11,12)(6,15,16,8)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,36)(34,40)(35,43)(37,47)(38,44)(39,45)(41,46)(42,48);(1,5,7,2)(3,8,11,15)(4,13,14,9)(6,12,16,10)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,35)(34,41)(36,43)(37,39)(38,42)(40,46)(44,48)(45,47);(1,6)(2,10)(3,9)(4,8)(5,12)(7,16)(11,13)(14,15)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(33,37,38,41)(34,35,39,42)(36,46,44,47)(40,48,45,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,d^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, 8537992);