Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 | |
1953 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,8)(7,10,12,11)(13,14,17,20)(15,25,26,16)(18,24,19,23)(21,22,28,27);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(14,20)(15,22)(16,28)(18,19)(21,25)(26,27);(5,8,9,6)(7,11,12,10)(13,15,19,22,17,26,18,27)(14,21,24,16,20,28,23,25);(7,12)(10,11)(13,16,18,21,17,25,19,28)(14,22,23,15,20,27,24,26);(1,3,4,2)(14,20)(16,25)(21,28)(23,24)Polycyclic presentation:
2^9|a^2=ghi,b^2=fi,c^2=h,d^2=g,e^2=fi,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=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72314130 |
| maximal normal subgroup of | #72314130 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511450);