#9719151

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

3939 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)(3,6)(4,5)(7,8)(9,13)(10,16)(11,14)(12,15)(17,18)(19,26)(20,22)(21,25)(23,27)(24,31)(28,32)(29,30)(33,34,40,43)(35,44,37,38)(36,45,46,41)(39,47,48,42);(1,3,7,10)(2,6,8,16)(4,11,13,15)(5,14,9,12)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,27)(33,35,40,37)(34,38,43,44)(36,42,46,47)(39,45,48,41);(2,8)(3,10)(4,13)(12,14)(17,20,23,21)(18,22,27,25)(19,28,31,29)(24,30,26,32)(33,36,40,46)(34,41,43,45)(35,42,37,47)(38,39,44,48);(1,4)(2,9)(3,11)(5,8)(6,12)(7,13)(10,15)(14,16)(17,21,23,20)(18,25,27,22)(19,29,31,28)(24,32,26,30)(33,37,40,35)(34,38,43,44)(36,42,46,47)(39,41,48,45);(1,5,7,9)(2,4,8,13)(3,12,10,14)(6,15,16,11)(17,19)(18,26)(20,28)(21,29)(22,32)(23,31)(24,27)(25,30)(33,38)(34,35)(36,39)(37,43)(40,44)(41,42)(45,47)(46,48);(1,6)(2,10)(3,8)(4,12)(5,11)(7,16)(9,15)(13,14)(17,22,23,25)(18,20,27,21)(19,30,31,32)(24,28,26,29)(33,39)(34,42)(35,45)(36,44)(37,41)(38,46)(40,48)(43,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9626347);