Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14,19,22)(15,17,21,18)(16,24,25,23)(20,26,28,27);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,14,19,21,17,22)(16,23,27,20,25,24,26,28);(7,11)(9,12)(13,16,17,26,19,25,18,27)(14,20,15,23,22,28,21,24);(7,11)(9,12)(16,25)(20,28)(23,24)(26,27);(13,17,19,18)(14,21,22,15)(16,26,25,27)(20,24,28,23)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=gi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72224791, #72224931, #72224939, … |
| maximal normal subgroup of | #72224791, #72224931, #72224939, … |
| maximal quotient of | #72224773 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364538);