Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23444 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2678 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)(5,6,9,14,10,13,21,27)(7,11,17,24,16,23,29,34)(8,12,20,26,19,25,32,35)(15,22,28,33,18,30,31,36);(5,7)(6,11)(8,18,19,15)(9,17)(10,16)(12,22,25,30)(13,23)(14,24)(20,31,32,28)(21,29)(26,33,35,36)(27,34);(2,4)(5,8)(6,12)(7,15)(9,20)(10,19)(11,22)(13,25)(14,26)(16,18)(17,28)(21,32)(23,30)(24,33)(27,35)(29,31)(34,36);(7,16)(11,23)(15,18)(17,29)(22,30)(24,34)(28,31)(33,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=di,g^b=gi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237694, #72237704, #72237708, … |
| maximal normal subgroup of | #72237694, #72237704, #72237708, … |
| maximal quotient of | #72314605, #72314659, C2×#357764 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264960);