Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395169 |
| 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 | C2×#2018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(14,16)(15,23)(17,18)(20,27)(21,22)(25,26);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,21,19,22,17,16)(15,20,25,28,26,27,23,24);(6,9)(10,12)(13,15)(14,20)(16,24)(17,23)(18,25)(19,26)(21,28)(22,27);(5,7,8,11)(6,10,9,12)(13,16,18,14,19,21,17,22)(15,24,25,20,26,28,23,27)Polycyclic presentation:
2^9|b^2=ei,d^2=gh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322459 |
| maximal normal subgroup of | #72322459 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321224);