#8607429

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

4116 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)(19,24)(20,21)(22,23)(25,30)(26,29)(27,31)(28,32)(34,40)(35,36)(37,45)(47,48);(1,2,6,3)(4,9,11,7)(5,8,13,10)(12,15,16,14)(17,18)(19,26)(20,21)(22,23)(24,29)(25,30)(27,28)(31,32)(33,34)(35,41)(36,38)(37,46)(39,40)(42,45)(43,47)(44,48);(4,11)(7,9)(12,16)(14,15)(18,23)(20,30)(24,26)(27,32)(33,35,39,36)(34,41,40,38)(37,44,45,43)(42,47,46,48);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,19)(18,24)(20,27)(21,31)(22,29)(23,26)(25,28)(30,32)(33,36,39,35)(34,41,40,38)(37,44,45,43)(42,48,46,47);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20)(18,25)(19,27)(21,23)(22,30)(24,28)(26,31)(29,32)(33,37)(34,42)(35,43)(36,44)(38,47)(39,45)(40,46)(41,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,25)(18,20,23,30)(19,28,29,31)(24,32,26,27)(33,38,39,41)(34,36,40,35)(37,47,45,48)(42,44,46,43)

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=g,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, 8514625);