Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 25165824 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1910 conjugacy classes of subgroups; only the 554 normal subgroups are shown.
This group has 1910 conjugacy classes of subgroups — too many to draw. Only its 554 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(7,11)(10,12)(13,14,17,16)(15,19,20,18)(21,22,24,26)(23,27,28,25);(6,9)(10,12)(13,15,17,20)(14,18,16,19)(21,22)(23,25)(24,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,16,17,14)(15,19,20,18)(21,23,24,28)(22,25,26,27);(1,2,3,4)(13,15)(14,19)(16,18)(17,20)(21,22)(23,27)(24,26)(25,28);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4371337);