#8821731

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

4169 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8728927);