Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1088 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)(5,6,9,15,10,14,22,29,11,16,23,30,8,12,21,25)(7,13,19,28,20,27,33,36,17,26,31,35,18,24,32,34);(2,4)(6,12,16,14)(13,24,26,27)(15,25,30,29)(28,34,35,36);(5,7,9,19,10,20,22,33,11,17,23,31,8,18,21,32)(6,13,15,28,14,27,29,36,16,26,30,35,12,24,25,34);(7,17)(13,26)(18,20)(19,31)(24,27)(28,35)(32,33)(34,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=i,f^2=h,h^2=i,b^a=be,e^a=ei,g^a=gi,d^c=di176 conjugacy classes — view the full character table.
| maximal subgroup of | #72314209, #72314220, #72314229, … |
| maximal normal subgroup of | #72314209, #72314220, #72314229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400170);