#9717750

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

4218 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)(19,23)(20,21)(22,24)(25,29)(26,27)(28,30)(31,32)(33,34,38,36)(35,41,43,39)(37,40,46,44)(42,48,45,47);(1,3,7,4)(2,5,8,11)(6,10,16,12)(9,13,14,15)(17,19,22,28)(18,23,24,30)(20,26,25,31)(21,27,29,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,48)(42,46)(44,47);(2,8)(3,4)(9,14)(10,12)(18,24)(20,25)(23,30)(26,31)(34,36)(39,41)(40,44)(47,48);(1,4,7,3)(2,5,8,11)(6,12,16,10)(9,13,14,15)(17,20,22,25)(18,21,24,29)(19,26,28,31)(23,27,30,32)(33,36,38,34)(35,41,43,39)(37,44,46,40)(42,48,45,47);(1,5)(2,3)(4,8)(6,15)(7,11)(9,12)(10,14)(13,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37,38,46)(34,40,36,44)(35,42,43,45)(39,47,41,48);(1,6,7,16)(2,9,8,14)(3,10,4,12)(5,13,11,15)(17,21)(18,25)(19,27)(20,24)(22,29)(23,31)(26,30)(28,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,48)(42,46)(44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=g,f^2=i,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, 9624946);