Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3003 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14,16,21)(15,22,23,18)(17,19,25,20)(24,27,28,26);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,23)(17,24)(19,26)(20,27)(21,22)(25,28);(2,4)(14,19)(17,25)(18,26)(20,21)(22,27)(24,28);(6,10)(7,12)(14,20)(15,23)(17,25)(18,26)(19,21)(22,27);(14,21)(18,22)(19,20)(26,27)Polycyclic presentation:
2^9|a^2=f,b^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72282783, #72283296, #72283487, … |
| maximal normal subgroup of | #72282783, #72283296, #72283487, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408248);