Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,31)(24,28)(25,29)(26,30)(27,32)(33,36)(34,35);(5,7)(6,8)(9,12)(10,11)(21,23)(22,27)(24,33)(25,32)(26,34)(28,30)(29,31)(35,36);(5,8,10,9)(6,11,12,7)(13,15,16,20)(14,17,18,19)(21,24,33,23,25,34,26,32)(22,28,36,31,29,35,30,27);(5,9,10,8)(6,11,12,7)(14,18)(17,19)(22,29)(23,32)(24,34)(30,36)Polycyclic presentation:
2^9|a^2=h,c^2=efg,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240759, #72240769, #72240780, … |
| maximal normal subgroup of | #72240759, #72240769, #72240780, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293711);