Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23892 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,23)(9,27)(10,15)(11,16)(12,20)(13,25)(14,24)(18,34)(19,29)(21,31)(22,30)(26,33)(28,35)(32,36);(1,3,4,2)(5,7)(6,12)(8,19)(9,21)(10,20)(11,22)(13,32)(14,29)(15,30)(16,31)(17,27)(18,24)(23,28)(25,35)(26,36)(33,34);(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,28)(14,16)(15,33)(17,29)(19,20)(21,36)(22,35)(23,27)(30,32)(31,34);(5,8)(6,14)(7,19)(9,25)(10,24)(11,26)(12,29)(13,16)(15,33)(17,28)(18,20)(21,35)(22,36)(23,27)(30,34)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72312219, #72312221, #72312227, … |
| maximal normal subgroup of | #72312219, #72312221, #72312227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271816);