Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14,19,23)(15,24,27,34)(16,21,28,36)(17,22,18,31)(20,32,35,26)(25,39,40,44)(29,37,30,38)(33,41,43,42);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(16,28)(21,36)(25,40)(29,30)(33,43)(37,38)(39,44)(41,42);(2,4)(13,15)(14,20)(16,25)(17,26)(18,32)(19,27)(21,33)(22,34)(23,35)(24,31)(28,40)(29,41)(30,42)(36,43)(37,39)(38,44);(6,10)(9,11)(13,16,19,28)(14,21,23,36)(15,25,27,40)(17,30,18,29)(20,33,35,43)(22,38,31,37)(24,39,34,44)(26,42,32,41);(2,4)(7,12)(9,11)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,29,28,30)(20,34,35,24)(21,37,36,38)(25,41,40,42)(33,39,43,44)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^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,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575227);