#8429522

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

4108 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8336718);