Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| 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 |
| prime radical | C22×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4828 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)(13,14,18,23)(15,20,16,21)(17,19,26,22)(24,27,25,28);(7,8)(10,11)(14,19)(17,26)(20,27)(21,28)(22,23)(24,25);(2,4)(5,7,9,8)(6,10,12,11)(13,15,18,16)(14,20,23,21)(17,24,26,25)(19,27,22,28);(5,8,9,7)(6,11,12,10)(13,16,18,15)(14,21,23,20)(17,25,26,24)(19,28,22,27);(6,12)(10,11)(15,16)(20,21)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72281271, #72284690, #72284814, … |
| maximal normal subgroup of | #72281271, #72284690, #72284814, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394678);