Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188764 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2283 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17,21,13)(8,22,23,27)(9,12)(10,15)(11,16)(14,25,30,24)(18,32)(19,29,20,26)(28,35)(31,34)(33,36);(6,12,16,15)(8,23)(13,26,17,29)(14,22,30,27)(18,33)(24,25)(28,36,31,32)(34,35);(5,7)(6,13)(8,18)(9,19)(10,20)(11,21)(12,26)(14,28)(15,29)(16,17)(22,36)(23,33)(24,34)(25,35)(27,32)(30,31);(2,4)(5,8,9,24,11,23,10,25)(6,14,15,22,16,30,12,27)(7,18,19,34,21,33,20,35)(13,28,29,36,17,31,26,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,d^b=di,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72257017, #72257156, #72257201, … |
| maximal normal subgroup of | #72257017, #72257156, #72257201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299573);