Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4446 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)(5,6,10,14,11,15,9,22)(7,16,19,12,18,27,23,26)(8,13,20,31,21,30,24,29)(17,32,35,25,33,36,34,28);(5,7)(6,12)(8,17)(9,23)(10,19)(11,18)(13,28)(14,27)(15,26)(16,22)(20,35)(21,33)(24,34)(25,30)(29,36)(31,32);(2,4)(5,8)(6,13)(7,17)(9,24)(10,20)(11,21)(12,25)(14,31)(15,30)(16,32)(18,33)(19,35)(22,29)(23,34)(26,28)(27,36);(7,18)(12,26)(16,27)(17,33)(19,23)(25,28)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^b=ei,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72236660, #72249684, #72249730, … |
| maximal normal subgroup of | #72236660, #72249684, #72249730, … |
| maximal quotient of | #72236693, #72314405 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264653);