Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220190 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6,11,18,12,16,10,17)(7,19,22,43,23,13,33,37)(8,24,28,40,29,14,27,41)(9,30,31,44,32,15,34,42)(20,35,25,38,48,45,52,53)(21,36,50,61,51,46,49,60)(26,39,56,63,57,54,58,64)(47,65,66,62,67,59,55,68);(1,2,4,3)(5,7,10,22,12,23,11,33)(6,13,17,37,16,19,18,43)(8,25,27,48,29,52,28,20)(9,21,31,49,32,51,34,50)(14,38,40,45,24,53,41,35)(15,36,42,60,30,46,44,61)(26,55,56,67,57,66,58,47)(39,62,63,65,54,68,64,59);(5,8,12,29)(6,14,16,24)(7,20,23,48)(9,26,32,57)(10,28,11,27)(13,35,19,45)(15,39,30,54)(17,41,18,40)(21,47,51,67)(22,52,33,25)(31,58,34,56)(36,59,46,65)(37,53,43,38)(42,64,44,63)(49,66,50,55)(60,68,61,62);(1,3,4,2)(5,9)(6,15)(7,21)(8,26)(10,31)(11,34)(12,32)(13,36)(14,39)(16,30)(17,42)(18,44)(19,46)(20,47)(22,49)(23,51)(24,54)(25,55)(27,56)(28,58)(29,57)(33,50)(35,59)(37,60)(38,62)(40,63)(41,64)(43,61)(45,65)(48,67)(52,66)(53,68)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=i,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269832);