Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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 |
| prime radical | #34738 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3289 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)(7,9)(8,10)(11,12)(13,14,19,23)(15,24,25,21)(16,20,26,28)(17,22,18,27);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,26,19,25,18,16)(14,20,27,21,23,28,22,24);(6,10)(7,11)(13,16,18,25,19,26,17,15)(14,21,22,20,23,24,27,28);(7,11)(9,12);(13,17,19,18)(14,22,23,27)(15,16,25,26)(20,21,28,24)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gh,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72281503, #72281515, #72282020, … |
| maximal normal subgroup of | #72281503, #72281515, #72282020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366963);