Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14,19,23)(15,20,26,36)(16,27,29,37)(17,22,18,32)(21,31,38,28)(24,39,42,44)(25,34,30,35)(33,40,43,41);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,24)(17,30)(18,25)(19,26)(21,33)(22,35)(23,36)(27,39)(28,41)(29,42)(31,40)(32,34)(37,44)(38,43);(2,4)(13,16,17,31,19,29,18,28)(14,21,32,37,23,38,22,27)(15,24,30,40,26,42,25,41)(20,33,34,44,36,43,35,39);(6,10)(9,11)(13,17,19,18)(14,22,23,32)(15,25,26,30)(16,28,29,31)(20,34,36,35)(21,37,38,27)(24,40,42,41)(33,39,43,44);(2,4)(7,12)(9,11)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589663);