#8393437

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

4160 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)(3,11)(4,12)(5,10)(6,8)(7,13)(9,15)(14,16)(18,23)(19,25)(26,30)(27,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,48)(42,46)(44,47);(1,3,6,13)(2,7,8,11)(4,10,15,14)(5,12,16,9)(17,18)(19,20)(21,26)(22,23)(24,25)(27,28)(29,30)(31,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,48)(42,46)(44,47);(2,8)(5,16)(7,11)(9,12)(21,29)(26,30)(27,32)(28,31)(34,38)(39,46)(40,43)(47,48);(1,4)(2,9)(3,10)(5,7)(6,15)(8,12)(11,16)(13,14)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,35,37,42)(34,39,38,46)(36,44,45,41)(40,48,43,47);(1,3)(2,7)(4,14)(5,12)(6,13)(8,11)(9,16)(10,15)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,36,37,45)(34,40,38,43)(35,41,42,44)(39,47,46,48);(1,5,6,16)(2,10,8,14)(3,12,13,9)(4,7,15,11)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,34)(35,39)(36,40)(37,38)(41,47)(42,46)(43,45)(44,48)

Polycyclic presentation:

2^9|b^2=hi,d^2=i,e^2=gh,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, 8300633);