Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1046 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)(7,18)(8,13)(9,14)(10,16)(11,17)(12,20)(15,25)(19,30)(21,28)(22,27)(23,32)(24,33)(26,31)(29,35)(34,36);(5,7,10,20,11,21,25,9)(6,12,15,18,17,14,16,28)(8,22,23,35,24,36,26,19)(13,29,31,27,33,30,32,34);(5,8,11,24)(6,13,17,33)(7,19,21,35)(9,22,20,36)(10,26,25,23)(12,27,14,34)(15,32,16,31)(18,29,28,30);(5,9,25,21,11,20,10,7)(6,14,15,28,17,12,16,18)(8,19,26,36,24,35,23,22)(13,30,31,34,33,29,32,27)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=i,d^2=gi,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279485 |
| maximal normal subgroup of | #72279485 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288373);