#8459500

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

4172 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8366696);