Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| 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 | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2308 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)(5,6,8,10)(7,11,12,9)(13,14,17,20)(15,22,16,19)(18,21,25,28)(23,27,24,26);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,17)(18,23)(20,22)(21,26)(24,25)(27,28);(2,4)(15,23)(16,24)(18,25)(19,26)(21,28)(22,27);(13,16)(14,19)(15,17)(18,24)(20,22)(21,26)(23,25)(27,28);(15,23)(16,24)(18,25)(19,26)(21,28)(22,27)Polycyclic presentation:
2^9|a^2=f,b^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72307082, #72307157, #72307303, … |
| maximal normal subgroup of | #72307082, #72307157, #72307303, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481632);