Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| 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×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14,19,22)(15,20,24,26)(16,23,25,27)(17,21,18,28);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,21)(17,27)(18,23)(19,24)(22,26)(25,28);(6,9)(7,11)(13,16,18,26,19,25,17,20)(14,15,21,23,22,24,28,27);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,21,22,28)(15,23,24,27)(16,20,25,26);(6,9)(7,11)(15,24)(16,25)(20,26)(23,27)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=g,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,e^d=eh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257752, #72263845, #72263877, … |
| maximal normal subgroup of | #72257752, #72263845, #72263877, … |
| maximal quotient of | #72264586 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513686);