#8696702

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

4109 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603898);