Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32875 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2842 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1464 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,17,19,12)(8,21,23,13)(9,15,10,14)(18,33,34,27)(20,28,26,32)(22,29,36,35)(24,30,25,31);(2,4)(5,7,10,20,11,19,9,26)(6,12,15,28,16,17,14,32)(8,22,25,34,23,36,24,18)(13,29,31,33,21,35,30,27);(1,2,3,4)(5,8,9,24,11,23,10,25)(6,13,14,30,16,21,15,31)(7,18,20,22,19,34,26,36)(12,27,28,29,17,33,32,35);(18,34)(22,36)(27,33)(29,35)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72210062, #72210063, #72210075, … |
| maximal normal subgroup of | #72210062, #72210063, #72210075, … |
| maximal quotient of | #72266464, C2×#392035, #72266601 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299231);