Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393989 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15)(8,12)(9,21)(10,14)(11,17)(13,22)(16,30)(18,29)(19,26)(20,35)(23,36)(24,33)(25,32)(27,34)(28,31);(5,7,9,17,10,18,22,33)(6,11,13,15,14,24,21,29)(8,16,20,32,19,31,34,36)(12,25,27,30,26,23,35,28);(5,8)(6,12)(7,16)(9,20)(10,19)(11,23)(13,27)(14,26)(15,28)(17,32)(18,31)(21,35)(22,34)(24,25)(29,30)(33,36);(1,2,3,4)(8,19)(12,26)(16,31)(20,34)(23,25)(27,35)(28,30)(32,36)Polycyclic presentation:
2^9|b^2=e,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257079, #72257101 |
| maximal normal subgroup of | #72257079, #72257101 |
| maximal quotient of | #72219667, #72257548 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318976);