Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4885 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)(7,10)(8,11)(9,12)(13,14)(15,26)(16,22)(17,23)(18,24)(19,25)(20,28)(21,27);(2,4)(7,8)(10,11)(13,15,19,16)(14,20,25,21)(17,27,18,28)(22,23,26,24);(5,7,9,8)(6,10,12,11)(13,15,19,16)(14,21,25,20)(17,27,18,28)(22,24,26,23);(5,8,9,7)(6,11,12,10)(13,16,19,15)(14,22,25,26)(17,27,18,28)(20,23,21,24);(6,12)(10,11)(13,17,19,18)(14,23,25,24)(15,27,16,28)(20,26,21,22)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72256862, #72256878, #72257082, … |
| maximal normal subgroup of | #72256862, #72256878, #72257082, … |
| maximal quotient of | #72264841 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394839);