Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394046 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2501 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(9,10,16,22)(11,23,28,39)(12,18,31,43)(13,32,34,44)(14,21,15,20)(17,27,40,26)(19,33,45,35)(24,47,51,65)(25,52,54,38)(29,57,59,62)(30,46,36,42)(37,56,64,50)(41,60,66,58)(48,53,61,55)(49,63,69,71)(67,70,72,68);(1,2)(3,6)(4,5)(7,8)(9,11,16,28)(10,17,22,40)(12,24,31,51)(13,25,34,54)(14,27,15,26)(18,37,43,64)(19,38,45,52)(20,39,21,23)(29,49,59,69)(30,56,36,50)(32,61,44,48)(33,55,35,53)(41,63,66,71)(42,65,46,47)(57,67,62,72)(58,70,60,68);(2,5)(6,8)(9,12)(10,18)(11,24)(13,29)(14,30)(15,36)(16,31)(17,37)(19,41)(20,42)(21,46)(22,43)(23,47)(25,49)(26,50)(27,56)(28,51)(32,62)(33,58)(34,59)(35,60)(38,63)(39,65)(40,64)(44,57)(45,66)(48,67)(52,71)(53,68)(54,69)(55,70)(61,72);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,25)(12,29)(14,35)(15,33)(16,34)(17,38)(18,41)(20,32)(21,44)(22,45)(23,48)(24,49)(26,55)(27,53)(28,54)(30,60)(31,59)(36,58)(37,63)(39,61)(40,52)(42,62)(43,66)(46,57)(47,67)(50,70)(51,69)(56,68)(64,71)(65,72)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311173);