Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,12,17,28)(8,20)(9,24)(10,15)(11,14)(13,22)(16,32,33,29)(18,34,19,27)(21,26,35,36)(23,31)(25,30);(5,7)(6,12,14,28)(8,21,23,35)(9,18)(10,19)(11,17)(13,29)(15,27,24,34)(16,22,33,25)(20,26)(30,32)(31,36);(1,2,3,4)(5,8,9,22,11,23,10,25)(6,13,15,20,14,30,24,31)(7,16,18,21,17,33,19,35)(12,26,27,32,28,36,34,29)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,h^a=hi,c^b=cf,d^b=di,g^b=gi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237659, #72237660, #72237667, … |
| maximal normal subgroup of | #72237659, #72237660, #72237667, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31179);