Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394321 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60444 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,13)(9,25)(10,14)(11,15)(12,18)(17,28)(19,33)(20,27)(21,36)(22,31)(23,29)(24,30)(26,34)(32,35);(5,7,9,18,11,19,10,20)(6,12,14,16,15,27,25,33)(8,21,22,17,24,34,23,35)(13,28,29,36,30,32,31,26);(5,8,11,24)(6,13,15,30)(7,17,19,35)(9,23,10,22)(12,26,27,36)(14,31,25,29)(16,32,33,28)(18,21,20,34);(2,4)Polycyclic presentation:
2^9|b^2=e,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285454, #72285470, #72285641, … |
| maximal normal subgroup of | #72285454, #72285470, #72285641, … |
| maximal quotient of | #72316759, #72317027 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301518);