#8188230

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
prime radical	#41466	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4037 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)(9,12)(10,13)(11,14)(15,16)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34,38,39)(35,37,41,46)(36,43,45,40)(42,48,47,44);(2,3)(7,9)(8,10)(14,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,34,38,39)(35,37,41,46)(36,40,45,43)(42,44,47,48);(5,13)(8,10)(11,16)(14,15)(18,21)(23,25)(24,28)(30,31)(34,39)(37,46)(40,43)(44,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,35,38,41)(34,37,39,46)(36,42,45,47)(40,44,43,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,36,38,45)(34,40,39,43)(35,42,41,47)(37,44,46,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(33,37)(34,41)(35,39)(36,44)(38,46)(40,47)(42,43)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,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,f^c=fgi

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, 8095426);