Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(15,19)(17,20)(21,22,26,30)(23,31,32,35)(24,29,25,28)(27,34,36,33);(1,3,4,2)(5,7,9,8)(6,10,12,11)(13,14,16,18)(15,20,19,17)(21,23)(22,27)(24,34)(25,33)(26,32)(28,35)(29,31)(30,36);(5,7,9,8)(6,11,12,10)(13,15,16,19)(14,17,18,20)(22,28)(24,25)(27,35)(29,30)(31,36)(33,34);(5,8,9,7)(6,11,12,10)(22,29)(23,32)(24,25)(27,35)(28,30)(31,36)Polycyclic presentation:
2^9|a^2=h,b^2=efhi,c^2=ef,d^2=efg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi,e^d=ei,f^d=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280821);