Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,18,20)(15,23,25,16)(17,21,19,22)(24,27,26,28);(6,10)(11,12)(13,15)(14,16)(17,26)(18,25)(19,24)(20,23)(21,27)(22,28);(1,2,3,4)(5,7,9,8)(6,11,10,12)(14,20)(15,24)(16,27)(17,19)(23,28)(25,26);(5,8,9,7)(6,12,10,11)(13,16)(14,15)(17,27)(18,23)(19,28)(20,25)(21,26)(22,24);(13,17)(14,21)(18,19)(20,22)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72285049, #72285116 |
| maximal normal subgroup of | #72285049, #72285116 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491720);