Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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 |
| prime radical | C2×#3055 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2407 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,23)(17,26)(18,21)(20,25)(22,27)(24,28);(1,2)(3,4)(6,10)(11,12)(13,15,18,17)(14,19,21,26)(16,22,24,25)(20,23,27,28);(2,4)(5,7,9,8)(6,11,10,12)(13,16,17,25,18,24,15,22)(14,20,19,23,21,27,26,28);(2,4)(5,8,9,7)(6,12,10,11);(16,24)(20,27)(22,25)(23,28)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72316280 |
| maximal normal subgroup of | #72316280 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351275);