#8416736

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

3999 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8323932);