#8393422

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300618);