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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2071 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,23,19,24,18,22)(15,21,25,28,26,27,16,20);(1,2)(3,4)(13,15)(14,20)(16,17)(18,25)(19,26)(21,22)(23,27)(24,28);(2,4)(6,9)(10,12)(13,15,17,25,19,26,18,16)(14,21,23,28,24,27,22,20);(2,4)(13,16,17,15,19,25,18,26)(14,20,23,21,24,28,22,27);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,24,23)(15,16,26,25)(20,21,28,27)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=f,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72257264, #72257653, #72263846, … |
| maximal normal subgroup of | #72257264, #72257653, #72263846, … |
| maximal quotient of | #72264369 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363739);