Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| 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 | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2270 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)(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,20)(17,19)(21,24)(22,25)(23,26)(27,28);(2,4)(15,21)(16,22)(18,24)(20,25)(23,27)(26,28);(5,7,8,12)(6,9,10,11)(13,15,16,21,17,23,22,27)(14,18,20,24,19,26,25,28);(13,15,22,27,17,23,16,21)(14,18,25,28,19,26,20,24);(14,19)(18,26)(20,25)(24,28)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239690, #72249399, #72249401, … |
| maximal normal subgroup of | #72239690, #72249399, #72249401, … |
| maximal quotient of | #72309075, C2×#6466006 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373202);