Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514000 |
| 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×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,12)(9,10)(13,14,17,22)(15,23,25,19)(16,20,18,21)(24,28,26,27);(2,3)(6,10)(11,12)(13,15)(14,19)(16,24)(17,25)(18,26)(20,27)(21,28)(22,23);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,17,18)(14,20,22,21)(15,24,25,26)(19,27,23,28);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,16)(14,20)(17,18)(21,22)(24,26)(27,28);(6,10)(11,12)(13,16,17,18)(14,21,22,20)(15,24,25,26)(19,28,23,27)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72281581, #72282998, #72283952, … |
| maximal normal subgroup of | #72281581, #72282998, #72283952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468758);