Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,22)(15,24,25,20)(16,21,29,36)(17,19,30,23)(26,38,39,35)(27,34,40,32)(28,37,31,33)(41,44,42,43);(5,6)(7,11)(8,10)(9,12)(14,19)(15,25)(16,28)(17,30)(20,34)(21,36)(22,23)(24,32)(26,41)(29,31)(39,42)(43,44);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,26)(17,27)(18,25)(19,32)(21,35)(22,24)(23,34)(28,42)(29,39)(30,40)(31,41)(33,43)(36,38)(37,44);(7,12)(9,11)(13,16,18,29)(14,21,22,36)(15,26,25,39)(17,31,30,28)(19,33,23,37)(20,35,24,38)(27,41,40,42)(32,43,34,44);(6,10)(9,11)(14,22)(15,25)(19,23)(21,36)(26,39)(27,40)(33,37)(41,42)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970001);