Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682500 |
| 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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,10)(11,12)(13,14)(15,21)(16,20)(17,19)(18,23)(22,28)(24,26)(25,27);(1,2)(3,4)(13,15,17,23)(14,18,19,21)(16,24,25,22)(20,28,27,26);(2,4)(5,6,10,12,8,11,7,9);(14,19)(18,21)(20,27)(26,28);(2,4)(6,11)(9,12)(13,16,17,25)(14,20,19,27)(15,22,23,24)(18,26,21,28)Polycyclic presentation:
2^9|b^2=f,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307490, #72307528, #72307555, … |
| maximal normal subgroup of | #72307490, #72307528, #72307555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593076);