Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,10)(8,11)(13,14)(15,21)(16,24)(17,19)(18,23)(20,25)(22,27)(26,28);(1,2)(3,4)(5,6,9,10)(7,8,12,11)(15,22)(18,26)(21,27)(23,28);(5,7)(6,11)(8,10)(9,12)(13,15,16,23,17,22,25,28)(14,18,20,21,19,26,24,27);(2,4)(6,10)(7,12)(13,14)(15,18)(16,20)(17,19)(21,23)(22,26)(24,25)(27,28);(5,8,9,11)(6,7,10,12)(14,19)(18,26)(20,24)(21,27)Polycyclic presentation:
2^9|b^2=hi,c^2=g,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285191, #72307733, #72307935, … |
| maximal normal subgroup of | #72285191, #72307733, #72307935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7092519);