Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,17)(16,18)(19,20)(21,22,25,28,27,23,26,24);(2,4)(13,15,16,19)(14,17,18,20)(21,23,27,22)(24,25,28,26);(5,7,9,8)(6,10,11,12)(21,24,26,23,27,28,25,22);(5,8,9,7)(6,10,11,12)(14,18)(17,20)(21,25,27,26)(22,24,23,28)Polycyclic presentation:
2^9|a^2=ei,b^2=h,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237235, #72237292, #72239610, … |
| maximal normal subgroup of | #72237235, #72237292, #72239610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 317796);