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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,12)(9,10)(14,19)(15,23)(16,17)(20,22)(21,28)(24,25);(1,2)(3,4)(6,10)(11,12)(14,20)(15,24)(16,17)(19,22)(21,28)(23,25);(5,7,9,8)(6,11,10,12)(13,14,16,19,18,22,17,20)(15,21,23,26,25,28,24,27);(2,4)(5,8,9,7)(6,12,10,11)(13,15,17,24,18,25,16,23)(14,21,20,27,22,28,19,26);(6,10)(11,12)(14,22)(19,20)(21,28)(26,27)Polycyclic presentation:
2^9|c^2=fg,d^2=fgi,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^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257604, #72257677, #72263869, … |
| maximal normal subgroup of | #72257604, #72257677, #72263869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469349);