#9937712

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

4018 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9844908);