Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3050 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)(8,10)(9,12)(13,14)(15,21)(16,24)(17,19)(18,27)(20,23)(22,26)(25,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(14,19)(15,16)(22,27)(23,25);(2,4)(13,15,17,16)(14,20,19,28)(18,25,26,23)(21,22,24,27);(6,10)(9,11)(13,16,17,15)(14,21,19,24)(18,25,26,23)(20,22,28,27);(2,4)(7,12)(9,11)(15,16)(20,28)(21,24)(23,25)Polycyclic presentation:
2^9|c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72256843, #72256855, #72256955, … |
| maximal normal subgroup of | #72256843, #72256855, #72256955, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575278);