#8552126

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

4182 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18,23,22)(19,24,26,21)(20,25,29,30)(27,31,32,28)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,3,6,11)(2,7,4,9)(5,10,14,13)(8,15,12,16)(17,19,23,26)(18,24,22,21)(20,27,29,32)(25,31,30,28)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(2,4)(7,9)(8,12)(15,16)(19,26)(21,24)(27,32)(28,31)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,21)(18,19)(20,28)(22,26)(23,24)(25,27)(29,31)(30,32)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,22,23,18)(19,21,26,24)(20,25,29,30)(27,31,32,28)(33,36,37,44)(34,39,40,43)(35,41,42,48)(38,45,46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8459322);