Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394746 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#400 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1166 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)(5,6,11,14,10,15,9,25)(7,16,19,29,20,28,18,12)(8,21,26,30,24,34,23,13)(17,27,33,31,22,35,32,36);(5,7,9,18,10,20,11,19)(6,12,14,16,15,29,25,28)(8,22,23,33,24,17,26,32)(13,27,21,31,30,35,34,36);(5,8,9,23,10,24,11,26)(6,13,14,21,15,30,25,34)(7,17,18,32,20,22,19,33)(12,27,16,31,29,35,28,36);(2,4)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307225, #72307337, #72307361 |
| maximal normal subgroup of | #72307225, #72307337, #72307361 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308899);