#9783121

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
prime radical	#48690	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4038 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9690317);