#8393445

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

4122 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,34)(35,39)(36,43)(37,38)(40,45)(41,48)(42,46)(44,47);(2,3)(4,11)(8,10)(12,16)(17,19,23,20)(18,24,25,26)(21,30,32,28)(22,29,27,31)(33,34)(35,39)(36,43)(37,38)(40,45)(41,48)(42,46)(44,47);(5,13)(8,10)(12,16)(14,15)(18,25)(22,27)(24,26)(29,31)(34,38)(39,46)(40,43)(47,48);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20,23,19)(18,26,25,24)(21,28,32,30)(22,31,27,29)(33,35,37,42)(34,39,38,46)(36,44,45,41)(40,48,43,47);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21)(18,27)(19,28)(20,30)(22,25)(23,32)(24,29)(26,31)(33,36,37,45)(34,40,38,43)(35,41,42,44)(39,47,46,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,27)(18,21,25,32)(19,29,20,31)(24,30,26,28)(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=gi,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, 8300641);