Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23990 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2496 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)(7,12)(8,19)(9,11)(10,15)(13,20)(14,23)(16,30)(17,26)(18,28)(21,25)(22,24)(27,31)(29,35)(32,33)(34,36);(6,11,15,14)(12,26,28,24)(13,25,29,19)(27,33,36,30);(5,7,10,18)(6,12,15,28)(8,16,21,32)(9,22,23,17)(11,24,14,26)(13,27,29,36)(19,33,25,30)(20,34,35,31);(5,8)(6,13,15,29)(7,16)(9,20)(10,21)(11,25,14,19)(12,27,28,36)(17,31)(18,32)(22,34)(23,35)(24,30,26,33)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^c=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261630, #72261661, #72306505, … |
| maximal normal subgroup of | #72261630, #72261661, #72306505, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271121);