Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,22)(17,25)(18,21)(19,24)(20,26)(27,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,17)(16,20)(18,25)(19,28)(21,23)(22,27)(24,26);(13,16)(17,27)(18,19)(20,28)(22,24)(23,26);(2,4)(13,17,18,23)(14,20,21,28)(15,24,25,22)(16,26,19,27);(6,10)(9,11)(14,21)(17,23)(22,24)(26,27)Polycyclic presentation:
2^9|b^2=fi,d^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=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72257319, #72257525, #72264140, … |
| maximal normal subgroup of | #72257319, #72257525, #72264140, … |
| maximal quotient of | #72265650 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421858);