Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417647 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,9)(10,11)(13,14,16,21,18,22,17,20)(15,19,24,27,25,26,23,28);(5,6,8,9)(7,11,12,10)(13,15)(14,19)(16,24)(17,23)(18,25)(20,28)(21,27)(22,26);(14,20)(15,23)(16,17)(19,26)(21,22)(24,25);(2,4)(5,7,8,12)(6,10,9,11)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321262 |
| maximal normal subgroup of | #72321262 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327340);