Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6,9,8)(7,11,12,10)(13,14,16,18)(15,17,19,20)(21,22,24,27)(23,28,25,26);(5,7,9,12)(6,10,8,11)(13,15,16,19)(14,17,18,20)(21,23,24,25)(22,26,27,28);(2,4)(5,8,9,6)(7,11,12,10)(15,19)(17,20);(7,12)(10,11)(22,26)(23,25)(27,28)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,c^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72315850, #72315892, #72315917, … |
| maximal normal subgroup of | #72315850, #72315892, #72315917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 347984);