#9713741

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

4270 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18)(19,29)(20,26)(21,27)(22,32)(23,25)(24,30)(28,31)(33,34)(35,43)(36,46)(37,40)(38,42)(39,41)(44,47)(45,48);(1,2)(3,4)(5,9)(6,7)(8,10)(11,12)(13,15)(14,16)(17,19,23,30)(18,24,25,29)(20,28,27,32)(21,31,26,22)(33,35,40,39)(34,41,37,43)(36,45,38,44)(42,48,46,47);(2,7)(3,8)(9,13)(11,16)(18,25)(20,27)(24,29)(28,32)(33,36,40,38)(34,42,37,46)(35,44,39,45)(41,47,43,48);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(17,20,23,27)(18,26,25,21)(19,28,30,32)(22,29,31,24)(33,37,40,34)(35,43,39,41)(36,42,38,46)(44,47,45,48);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,21,23,26)(18,27,25,20)(19,22,30,31)(24,28,29,32)(33,38,40,36)(34,42,37,46)(35,45,39,44)(41,47,43,48);(1,5)(2,9)(3,11)(4,12)(6,15)(7,13)(8,16)(10,14)(17,22)(18,28)(19,21)(20,29)(23,31)(24,27)(25,32)(26,30)(33,39,40,35)(34,43,37,41)(36,45,38,44)(42,48,46,47)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=h,e^2=g,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, 9620937);