Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1917 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)(5,6,11,14,10,15,9,24)(7,16,26,28,17,32,19,12)(8,20,25,31,23,13,22,30)(18,29,35,36,34,33,21,27);(5,7,9,19,10,17,11,26)(6,12,14,16,15,28,24,32)(8,21,22,34,23,35,25,18)(13,29,30,36,20,33,31,27);(2,4)(7,17)(9,11)(14,24)(16,32)(18,34)(22,25)(29,33)(30,31);(5,8,10,23)(6,13,15,20)(7,18,17,34)(9,25,11,22)(12,27,28,36)(14,31,24,30)(16,33,32,29)(19,35,26,21)Polycyclic presentation:
2^9|a^2=ef,b^2=e,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72228038 |
| maximal normal subgroup of | #72228038 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326391);