#8621997

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

4109 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8529193);