#8159234

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

4049 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)(3,9,4,7)(5,13,14,10)(11,15,12,16)(17,18)(19,21)(20,28)(22,23)(24,30)(25,26)(27,32)(29,31)(33,34,38,42)(35,43,45,39)(36,40,46,44)(37,47,48,41);(1,3,6,4)(2,7,8,9)(5,11,14,12)(10,15,13,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,35)(34,39)(36,41)(37,44)(38,45)(40,48)(42,43)(46,47);(2,8)(7,9)(10,13)(15,16)(18,23)(19,25)(24,28)(27,31)(33,34,38,42)(35,39,45,43)(36,40,46,44)(37,41,48,47);(1,2)(3,9)(4,7)(5,10)(6,8)(11,16)(12,15)(13,14)(17,18)(19,26)(20,28)(21,25)(22,23)(24,30)(27,29)(31,32)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,45)(47,48);(1,4,6,3)(2,9,8,7)(5,12,14,11)(10,16,13,15)(17,20,22,30)(18,24,23,28)(19,27,25,31)(21,29,26,32)(33,36)(34,40)(35,41)(37,43)(38,46)(39,48)(42,44)(45,47);(1,5)(2,10)(3,11)(4,12)(6,14)(7,15)(8,13)(9,16)(17,21,22,26)(18,25,23,19)(20,29,30,32)(24,31,28,27)(33,37,38,48)(34,41,42,47)(35,44,45,40)(36,43,46,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066430);