Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | #24425 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2153 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)(7,10)(8,9)(11,12)(13,14,18,24)(15,25,28,20)(16,22,29,38)(17,23,19,21)(26,39,40,34)(27,35,31,33)(30,36,32,37)(41,43,42,44);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,26)(17,31)(18,28)(19,27)(21,33)(22,34)(23,35)(24,25)(29,40)(30,42)(32,41)(36,43)(37,44)(38,39);(2,4)(14,21)(16,29)(17,19)(20,33)(22,37)(23,24)(25,35)(26,40)(27,31)(34,44)(36,38)(39,43);(2,4)(13,16)(14,22)(15,26)(17,32)(18,29)(19,30)(20,34)(21,36)(23,37)(24,38)(25,39)(27,42)(28,40)(31,41)(33,43)(35,44);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,24,21)(15,27,28,31)(16,30,29,32)(20,33,25,35)(22,36,38,37)(26,41,40,42)(34,44,39,43)Polycyclic presentation:
2^9|a^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347476);