Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232351 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #34575 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1214 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1214 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,15)(8,12)(9,13)(10,14)(11,18)(16,19)(17,20)(21,22)(23,26)(24,27)(25,28)(29,34)(30,33)(31,35)(32,36);(1,2)(3,4)(5,7)(6,11)(8,17)(9,18)(10,16)(12,19)(13,15)(14,20)(21,23)(22,26)(24,29)(25,30)(27,34)(28,33)(31,36)(32,35);(1,3,4,2)(6,12)(7,16)(8,10)(13,14)(17,18)(19,20)(21,24)(22,27)(23,29)(25,31)(26,32)(28,35)(30,36)(33,34);(5,8,9,10)(6,12,13,14)(7,17,18,16)(11,19,15,20)(24,31)(27,35)(29,36)(32,34)Polycyclic presentation:
2^9|c^2=eh,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dhi,f^c=fh,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145662);