Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,22)(16,25)(17,19)(18,21)(20,24)(23,27)(26,28);(5,7)(6,9)(8,12)(10,11)(14,19)(15,23)(16,18)(20,27)(22,28)(24,26);(2,4)(13,15,17,24)(14,20,19,22)(16,23,18,26)(21,27,25,28);(2,4)(6,10)(7,12)(13,14)(15,20)(16,25)(17,19)(18,21)(22,24)(23,28)(26,27);(13,16,17,18)(14,21,19,25)(15,23,24,26)(20,27,22,28)Polycyclic presentation:
2^9|a^2=gi,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307372, #72307760, #72307827, … |
| maximal normal subgroup of | #72307372, #72307760, #72307827, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410362);