Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,8)(7,11,12,10)(13,14,16,18)(15,17,19,20)(21,22,25,29)(23,31,32,27)(24,33,35,28)(26,30,34,36);(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,17,18,20)(21,23)(22,27)(24,34)(25,32)(26,35)(28,36)(29,31)(30,33);(5,6,9,8)(7,10,12,11)(14,18)(17,20)(21,24,34,23,25,35,26,32)(22,28,36,27,29,33,30,31);(2,4)(5,8,9,6)(7,10,12,11)(23,32)(24,35)(27,31)(28,33)Polycyclic presentation:
2^9|a^2=efi,b^2=fi,c^2=eg,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351273);