Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682532 |
| 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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2081 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)(7,10)(8,9)(11,12)(13,14,18,22)(15,23,26,20)(16,17,21,19)(24,25,28,27);(1,2)(3,4)(6,9)(7,11)(13,15,18,26)(14,20,22,23)(16,24,21,28)(17,27,19,25);(2,4)(13,16,19,14,18,21,17,22)(15,24,25,20,26,28,27,23);(6,9)(10,12)(13,17,18,19)(14,16,22,21)(15,25,26,27)(20,28,23,24);(2,4)(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,21,22,16)(15,25,26,27)(20,24,23,28)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=i,e^2=f,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=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72307744, #72307888, #72308001, … |
| maximal normal subgroup of | #72307744, #72307888, #72308001, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592499);