Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,10)(7,11,8,12)(13,14,18,22)(15,24,25,17)(16,23,19,21)(20,28,26,27);(1,2)(3,4)(6,10)(11,12)(13,15)(14,17)(16,26)(18,25)(19,20)(21,27)(22,24)(23,28);(13,16)(15,25)(17,27)(18,19)(21,23)(24,28);(2,4)(5,7,9,8)(6,11,10,12)(13,17)(14,20)(15,21)(16,27)(18,24)(19,28)(22,26)(23,25);(5,8,9,7)(6,11,10,12)(13,16,18,19)(14,21,22,23)(15,26,25,20)(17,27,24,28)Polycyclic presentation:
2^9|a^2=f,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72263925, #72263942, #72263976, … |
| maximal normal subgroup of | #72263925, #72263942, #72263976, … |
| maximal quotient of | #72264172, #72264670 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429189);