Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1970 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,22)(15,18,21,17)(16,20,26,28)(23,24,27,25);(1,2,3,4)(5,7,8,11)(6,9,10,12)(13,15,17,14,19,21,18,22)(16,20,24,23,26,28,25,27);(6,10)(9,12)(13,16)(14,20)(15,23)(17,25)(18,24)(19,26)(21,27)(22,28);(2,4)(14,15)(17,18)(20,23)(21,22)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=i,b^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=dei,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328246);