Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,12)(9,10)(13,14)(15,23)(16,19)(17,27)(18,22)(20,25)(21,26)(24,28);(1,2)(3,4)(6,10)(11,12)(15,24)(16,25)(17,26)(19,20)(21,27)(23,28);(5,7,9,8)(6,11,10,12)(13,15,17,16,18,25,26,24)(14,19,21,23,22,28,27,20);(2,4)(5,8,9,7)(6,12,10,11)(13,16,26,15,18,24,17,25)(14,20,27,28,22,23,21,19);(6,10)(11,12)(15,25)(16,24)(19,28)(20,23)Polycyclic presentation:
2^9|c^2=fg,d^2=fgi,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=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72284211, #72284331, #72284407 |
| maximal normal subgroup of | #72284211, #72284331, #72284407 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469381);