Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 | C2×#3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2906 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)(7,12)(8,10)(9,11)(13,14,18,23)(15,24,16,20)(17,22,19,27)(21,25,28,26);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,16)(14,20,23,24)(17,26,19,25)(21,22,28,27);(2,4)(13,16,18,15)(14,21,23,28)(17,26,19,25)(20,22,24,27);(1,2)(3,4)(5,8,9,7)(6,10,11,12);(2,4)(13,17,18,19)(14,22,23,27)(15,25,16,26)(20,21,24,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72263980, #72263994, #72264050, … |
| maximal normal subgroup of | #72263980, #72263994, #72264050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484133);