Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #397 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,15,18,11)(8,19,22,30)(9,14)(10,13)(12,21,28,23)(16,32)(17,31,35,26)(20,36)(24,29)(25,33)(27,34);(1,2)(3,4)(5,7)(6,11)(8,20,22,16)(9,17)(10,18)(12,27,28,25)(13,15)(14,26)(19,32,30,36)(21,34,23,33)(24,35)(29,31);(2,4)(5,8)(6,12)(7,16)(9,23)(10,22)(11,25)(13,28)(14,30)(15,27)(17,34)(18,20)(19,29)(21,24)(26,32)(31,36)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fi,g^b=gi,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237647, #72237655, #72237671, … |
| maximal normal subgroup of | #72237647, #72237655, #72237671, … |
| maximal quotient of | #72249708, #72249724, C2×#123616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30812);