Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,12,17,20)(8,13)(9,11)(10,14)(15,16,18,19);(1,2)(3,4)(6,11)(8,14)(9,13)(12,18)(15,20)(16,19);(2,4)(5,7)(6,12)(8,15)(9,16)(10,17)(11,18)(13,19)(14,20);(5,8,10,11)(6,13,14,9)(7,15,17,18)(12,19,20,16)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214076, #72216059, #72216071, … |
| maximal normal subgroup of | #72214076, #72216059, #72216071, … |
| maximal quotient of | #72233803, #72233818, C2×#449496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356692);