Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,12)(9,10)(13,14)(15,20)(16,27)(17,22)(18,23)(19,24)(21,26)(25,28);(1,2)(3,4)(6,10)(11,12)(13,15,19,26)(14,20,24,21)(16,17,25,18)(22,28,23,27);(5,7,9,8)(6,11,10,12)(13,16,17,26,19,25,18,15)(14,21,23,27,24,20,22,28);(2,4)(13,16,18,15,19,25,17,26)(14,21,22,28,24,20,23,27);(5,8,9,7)(6,11,10,12)(13,17,19,18)(14,22,24,23)(15,25,26,16)(20,28,21,27)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=gi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257120, #72257130, #72257166, … |
| maximal normal subgroup of | #72257120, #72257130, #72257166, … |
| maximal quotient of | #72264155, #72308419, #72320275 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7471513);