#8898894

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

4133 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,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, 8806090);