Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24451 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1929 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,25)(17,21)(18,26)(19,22)(23,27)(24,28);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,18)(17,23)(19,24)(21,27)(22,28)(25,26);(5,7,8,11)(6,10,9,12)(13,16,18,15,19,23,17,24)(14,20,21,25,22,28,26,27);(2,4)(13,16,17,24,19,23,18,15)(14,20,26,27,22,28,21,25);(6,9)(10,12)(13,17,19,18)(14,21,22,26)(15,23,24,16)(20,27,28,25)Polycyclic presentation:
2^9|c^2=fg,d^2=gi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72282853, #72282899 |
| maximal normal subgroup of | #72282853, #72282899 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468422);