Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2503 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)(7,11)(10,12)(13,14,17,16,18,20,25,27)(15,21,24,22,23,19,28,26);(6,9)(7,11)(13,14)(15,21)(16,25)(17,27)(18,20)(19,23)(22,28)(24,26);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,26)(17,24)(18,23)(20,21)(22,27)(25,28);(6,9)(10,12)(13,16,25,14,18,27,17,20)(15,22,28,21,23,26,24,19);(5,7,8,11)(6,10,9,12)(14,20)(15,23)(16,27)(24,28)Polycyclic presentation:
2^9|a^2=f,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281810, #72282267, #72283280, … |
| maximal normal subgroup of | #72281810, #72282267, #72283280, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053294);