Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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 | C2×#3058 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14)(15,16)(17,22)(18,28)(19,23)(20,25)(21,24)(26,27);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,25,19,26,17,21)(14,20,22,16,23,24,28,27);(7,11)(9,12)(13,16,18,24,19,27,17,20)(14,21,22,15,23,25,28,26);(7,11)(9,12)(13,14)(15,24)(16,21)(17,28)(18,22)(19,23)(20,26)(25,27);(13,17,19,18)(14,22,23,28)(15,21,26,25)(16,24,27,20)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237227, #72237229, #72239652, … |
| maximal normal subgroup of | #72237227, #72237229, #72239652, … |
| maximal quotient of | #72237235, #72307703 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364568);