Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,22)(17,23)(18,20)(19,24)(25,27)(26,28);(2,4)(14,20)(15,16)(17,19)(21,28)(22,27)(25,26);(6,9)(10,12)(13,15)(14,21)(16,17)(18,25)(19,26)(20,27)(22,23)(24,28);(13,16,18,26)(14,22,20,28)(15,17,25,19)(21,23,27,24);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,20,24)(15,16,25,26)(21,22,27,28)Polycyclic presentation:
2^9|a^2=fi,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72308736, #72308822 |
| maximal normal subgroup of | #72308736, #72308822 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361425);