Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417534 |
| 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 | |
3413 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(6,10)(8,16)(11,19)(12,17)(13,20)(14,18);(5,6)(7,13)(8,17)(9,12)(10,16)(11,14)(15,19)(18,20);(8,16)(10,17)(13,19)(14,18);(2,4)(5,7)(6,11)(8,18)(9,15)(10,19)(12,20)(13,17)(14,16)Polycyclic presentation:
2^9|a^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72227999, #72228003, #72228004, … |
| maximal normal subgroup of | #72227999, #72228003, #72228004, … |
| maximal quotient of | #72256364, #72309784 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356695);