#9206519

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

4124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9113715);