Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2474 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)(7,11)(8,9)(10,12)(13,14,17,23,18,22,26,19)(15,20,24,28,16,21,25,27);(2,4)(6,9)(10,11)(14,19)(15,24)(16,25)(17,26)(22,23)(27,28);(5,7,8,12)(6,10,9,11)(13,15,17,24,18,16,26,25)(14,20,23,28,22,21,19,27);(13,16,26,24,18,15,17,25)(14,21,19,28,22,20,23,27);(14,22)(19,23)(20,21)(27,28)Polycyclic presentation:
2^9|a^2=fg,c^2=gh,d^2=hi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239657, #72249401, #72249419, … |
| maximal normal subgroup of | #72239657, #72249401, #72249419, … |
| maximal quotient of | C2×#6466187 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373383);