#9947213

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

4096 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)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,31)(30,32)(33,34)(35,36)(37,41)(38,46)(39,40)(42,47)(43,48)(44,45);(4,11)(8,14)(10,15)(12,16)(17,19,22,26)(18,23,24,21)(20,27,29,32)(25,28,31,30)(34,40)(36,41)(42,46)(44,48);(2,7)(3,9)(8,14)(10,15)(18,24)(19,26)(25,31)(27,32)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,19)(18,21)(20,27)(22,26)(23,24)(25,30)(28,31)(29,32)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32)(33,37,39,35)(34,41,40,36)(38,43,47,45)(42,44,46,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21)(18,19)(20,28)(22,23)(24,26)(25,32)(27,31)(29,30)(33,38,39,47)(34,42,40,46)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=hi,e^2=g,f^2=g,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, 9854409);