#9713739

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

4042 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18)(19,28)(20,26)(21,27)(22,24)(23,25)(29,32)(30,31)(34,37)(39,41)(40,45)(46,47);(1,2)(3,4)(5,9)(6,7)(8,10)(11,12)(13,15)(14,16)(17,19,23,22)(18,24,25,28)(20,29,27,31)(21,32,26,30)(36,44)(40,45)(42,48)(46,47);(2,7)(3,8)(9,13)(11,16)(18,25)(20,27)(24,28)(29,31)(33,34,38,37)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(17,20,23,27)(18,26,25,21)(19,29,22,31)(24,30,28,32)(33,35,38,43)(34,39,37,41)(36,42,44,48)(40,46,45,47);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,21,23,26)(18,27,25,20)(19,30,22,32)(24,29,28,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,5)(2,9)(3,11)(4,12)(6,15)(7,13)(8,16)(10,14)(17,22,23,19)(18,28,25,24)(20,31,27,29)(21,30,26,32)(33,37,38,34)(35,41,43,39)(36,45,44,40)(42,47,48,46)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=h,e^2=g,f^2=hi,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, 9620935);