#8549589

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

4096 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=g,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8456785);