Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61569 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#546 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,23)(17,26)(18,21)(20,22)(24,28)(25,27);(5,7)(6,9)(8,12)(10,11)(13,15,18,17)(14,19,21,26)(16,24,25,22)(20,23,28,27);(13,16)(14,20)(15,22)(17,24)(18,25)(19,27)(21,28)(23,26);(7,12)(9,11)(13,15,18,17)(14,19,21,26)(16,22,25,24)(20,27,28,23);(6,10)(9,11)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,g^c=gi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72315552, #72315558, #72315570 |
| maximal normal subgroup of | #72315552, #72315558, #72315570 |
| maximal quotient of | #72316288, #72316411 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293209);