Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 | C2×#3150 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2250 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)(7,11)(8,9)(10,12)(13,14)(15,20)(16,24)(17,21)(18,25)(19,23)(22,27)(26,28);(1,2)(3,4)(6,9)(10,11)(13,15,19,25)(14,20,23,18)(16,26,27,21)(17,24,28,22);(13,16,20,28,19,27,18,17)(14,21,25,22,23,26,15,24);(2,4)(13,17,18,27,19,28,20,16)(14,22,15,21,23,24,25,26);(5,7,8,12)(6,10,9,11)(16,27)(17,28)(21,26)(22,24)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=g,e^2=f,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=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72263899, #72263965, #72263980 |
| maximal normal subgroup of | #72263899, #72263965, #72263980 |
| maximal quotient of | #72264877 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469807);