Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9495 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C24 | C2×C42 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
1588 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 1588 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,21,23,18)(16,19)(17,20)(22,25,28,27)(24,26);(5,7)(6,9,10,11)(8,12)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(1,2,3,4)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efh,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72251433, #72251440, #72251448, … |
| maximal normal subgroup of | #72251433, #72251440, #72251448, … |
| maximal quotient of | #72255805, #72255983 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17598);