Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2024 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)(7,12)(8,10)(9,11)(15,21)(16,24)(18,22)(20,25)(23,26)(27,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,14,17,19)(15,22,23,28)(16,25,24,20)(18,26,27,21);(5,8,9,7)(6,10,11,12)(13,15,17,23)(14,18,19,27)(16,26,24,21)(20,28,25,22);(2,4)(13,15)(14,18)(16,26)(17,23)(19,27)(20,28)(21,24)(22,25);(6,11)(10,12)(14,19)(15,23)(20,25)(21,26)Polycyclic presentation:
2^9|b^2=fi,c^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=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72257349, #72257644, #72257761, … |
| maximal normal subgroup of | #72257349, #72257644, #72257761, … |
| maximal quotient of | #72264407 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7423164);