Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,3,4)(5,6)(7,17)(8,13)(9,26)(10,29)(11,16)(12,20)(14,27)(15,32)(18,44)(19,34)(21,40)(22,42)(23,52)(24,55)(25,38)(28,41)(30,35)(31,39)(33,51)(36,53)(37,57)(43,47)(45,66)(46,59)(48,61)(49,60)(50,58)(54,63)(56,62)(64,68)(65,67);(5,7)(6,12)(8,22)(9,19)(10,30)(11,21)(13,33)(14,34)(15,40)(16,35)(17,29)(18,25)(20,32)(23,50)(24,46)(26,43)(27,49)(28,48)(31,47)(36,58)(37,44)(38,59)(39,61)(41,60)(42,55)(45,54)(51,57)(52,64)(53,67)(56,68)(62,66)(63,65);(2,4)(5,8,11,25)(6,13,16,38)(7,18)(9,23,28,54)(10,24,32,57)(12,33)(14,36,39,62)(15,37,29,55)(17,42)(19,45)(20,46)(21,22)(26,52,41,63)(27,53,31,56)(30,51)(34,58)(35,59)(40,44)(43,64)(47,67)(48,50)(49,68)(60,65)(61,66);(5,9,11,28)(6,14,16,39)(7,19,21,48)(8,23,25,54)(10,31,32,27)(12,34,35,61)(13,36,38,62)(15,41,29,26)(17,43,40,60)(18,45,22,50)(20,49,30,47)(24,56,57,53)(33,58,59,66)(37,63,55,52)(42,64,44,65)(46,68,51,67)Polycyclic presentation:
2^9|a^2=h,c^2=fhi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281297);