Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | |
2948 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)(13,14,18,15)(16,24,26,25)(17,22,19,21)(20,28,23,27);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,25)(17,21)(19,22)(20,28)(23,27)(24,26);(13,16)(14,20)(15,23)(17,27)(18,26)(19,28)(21,24)(22,25);(2,4)(14,21)(15,22)(16,26)(17,19)(20,25)(23,24);(6,10)(9,11)(13,17,18,19)(14,21,15,22)(16,27,26,28)(20,24,23,25)Polycyclic presentation:
2^9|a^2=i,b^2=fi,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=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72282156, #72283956, #72284307, … |
| maximal normal subgroup of | #72282156, #72283956, #72284307, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422001);