Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1964 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)(7,11)(8,9)(10,12)(14,20)(15,24)(16,21)(17,19)(22,26)(25,27);(2,4)(6,9)(10,11)(13,14,18,20)(15,25,26,16)(17,23,19,28)(21,22,27,24);(13,15,19,24,18,26,17,22)(14,21,23,25,20,27,28,16);(1,2,3,4)(13,16,19,21,18,25,17,27)(14,22,23,15,20,24,28,26);(5,7,8,12)(6,10,9,11)(13,17,18,19)(14,23,20,28)(15,22,26,24)(16,21,25,27)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72314158, #72314252 |
| maximal normal subgroup of | #72314158, #72314252 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521151);