#8210226

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3854 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8117422);