Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,12)(7,11,8,10)(13,14)(15,22)(16,19)(17,21)(18,26)(20,24)(23,25)(27,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,14)(15,23)(16,20)(17,21)(18,27)(19,24)(22,25)(26,28);(2,4)(5,7,9,8)(6,11,12,10)(13,15,24,25,17,27,16,26)(14,18,19,28,21,23,20,22);(2,4)(5,8,9,7)(6,10,12,11)(13,14)(15,18)(16,20)(17,21)(19,24)(22,26)(23,27)(25,28);(13,16,17,24)(14,19,21,20)(15,25,27,26)(18,22,23,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249971, #72250017, #72256834, … |
| maximal normal subgroup of | #72249971, #72250017, #72256834, … |
| maximal quotient of | #72265542 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351912);