#9726099

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

4120 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)(3,5)(4,11)(6,14)(7,8)(9,16)(10,15)(12,13)(17,18)(19,29)(20,26)(21,27)(22,32)(23,25)(24,30)(28,31)(35,37)(39,41)(43,44)(46,47);(1,3,7,4)(2,5,8,11)(6,10,16,12)(9,13,14,15)(17,19,23,30)(18,24,25,29)(20,28,27,32)(21,31,26,22)(33,34,38,42)(35,39,37,41)(36,40,45,48)(43,46,44,47);(2,8)(3,4)(9,14)(10,12)(18,25)(20,27)(24,29)(28,32)(33,35,38,37)(34,39,42,41)(36,44,45,43)(40,47,48,46);(1,4,7,3)(2,5,8,11)(6,12,16,10)(9,13,14,15)(17,20,23,27)(18,26,25,21)(19,28,30,32)(22,29,31,24)(33,36,38,45)(34,40,42,48)(35,43,37,44)(39,46,41,47);(1,5)(2,3)(4,8)(6,15)(7,11)(9,12)(10,14)(13,16)(17,21,23,26)(18,27,25,20)(19,22,30,31)(24,28,29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,6,7,16)(2,9,8,14)(3,10,4,12)(5,13,11,15)(17,22)(18,28)(19,21)(20,29)(23,31)(24,27)(25,32)(26,30)(33,37,38,35)(34,41,42,39)(36,44,45,43)(40,47,48,46)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=h,e^2=g,f^2=gh,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, 9633295);