Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
2814 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)(15,20)(16,28)(17,23)(18,33)(19,24)(21,30)(22,26)(25,37)(27,36)(29,38)(31,39)(32,35)(34,40)(41,43)(42,44);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,25)(17,32)(18,26)(19,27)(21,34)(22,33)(23,35)(24,36)(28,37)(29,42)(30,40)(31,41)(38,44)(39,43);(2,4)(13,16,19,31)(14,21,24,38)(15,25,27,41)(17,30,18,29)(20,34,36,44)(22,37,35,43)(23,39,33,28)(26,42,32,40);(6,10)(7,12)(13,15)(14,22)(16,25)(17,26)(18,32)(19,27)(20,33)(21,37)(23,36)(24,35)(28,34)(29,40)(30,42)(31,41)(38,43)(39,44);(13,17,19,18)(14,23,24,33)(15,26,27,32)(16,29,31,30)(20,22,36,35)(21,28,38,39)(25,40,41,42)(34,43,44,37)Polycyclic presentation:
2^9|c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407913);