Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #398102 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1461 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)(7,11)(8,12,20,26)(9,13)(10,14)(15,22,18,30)(16,23)(17,24)(19,25,32,35)(21,27)(28,33,31,36)(29,34);(5,7)(6,11)(8,18,20,15)(9,16)(10,17)(12,22,26,30)(13,23)(14,24)(19,31,32,28)(21,29)(25,33,35,36)(27,34);(5,8)(6,12,14,26)(7,15)(9,19)(10,20)(11,22,24,30)(13,25,27,35)(16,28)(17,18)(21,32)(23,33,34,36)(29,31);(2,4)(5,9,10,21)(6,13,14,27)(7,16,17,29)(8,19,20,32)(11,23,24,34)(12,25,26,35)(15,28,18,31)(22,33,30,36)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,d^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320194 |
| maximal normal subgroup of | #72320194 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305299);