Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| 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)(7,17,21,12)(8,22)(9,16)(10,15)(11,14)(13,25)(18,34,35,30)(19,28,20,29)(23,33,36,27)(24,32)(26,31);(5,7)(6,12)(8,23,24,36)(9,19)(10,20)(11,21)(13,30,31,34)(14,17)(15,28)(16,29)(18,25,35,26)(22,33,32,27);(1,2,3,4)(5,8,9,25,11,24,10,26)(6,13,15,22,14,31,16,32)(7,18,19,23,21,35,20,36)(12,27,28,30,17,33,29,34)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,f^a=fi,c^b=cf,g^b=gi,d^c=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237647, #72237660, #72237668, … |
| maximal normal subgroup of | #72237647, #72237660, #72237668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31187);