Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144398 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10337 |
| commutator, Frattini, cosocle | C23×D4 | C23 |
| prime radical | #23895 | C2 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
818 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 818 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,15,21)(4,16)(5,10)(6,11)(7,14,22,30)(8,17)(9,19)(13,26,29,20)(18,23)(24,31)(25,27,32,28);(1,3)(2,7)(4,13)(5,14)(6,15)(8,20)(9,12)(10,21)(11,22)(16,25)(17,27)(18,29)(19,30)(23,32)(24,26)(28,31);(1,4)(2,8)(3,13)(5,17)(6,18)(7,20)(9,23)(10,16)(11,24)(12,25)(14,28)(15,29)(19,31)(21,32)(22,26)(27,30)Polycyclic presentation:
2^9|a^2=g,b^a=bd,c^a=ce,d^a=di,f^a=fgh,c^b=cf,e^b=eh,g^b=gi,d^c=dg,h^c=hi,f^d=fi,f^e=fi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72210453, #72213523, #72237632, … |
| maximal normal subgroup of | #72210453, #72213523, #72237632, … |
| maximal quotient of | #72258532, #72266686, C2×#144398 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51594);