Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,26)(16,32)(17,23)(18,24)(19,37)(20,25)(21,30)(22,28)(27,44)(29,39)(31,40)(33,38)(34,42)(35,41)(36,43);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,29,20,31,19,30)(14,21,37,40,25,39,24,26)(16,27,35,34,33,17,28,36)(22,23,38,42,41,44,32,43);(7,11)(9,12)(13,16,18,35,20,33,19,28)(14,22,37,38,25,41,24,32)(15,27,29,34,31,17,30,36)(21,23,40,42,39,44,26,43);(7,11)(9,12)(13,17,20,27)(14,23,25,44)(15,28,31,35)(16,30,33,29)(18,36,19,34)(21,38,39,32)(22,40,41,26)(24,43,37,42);(6,10)(9,12)(13,18,20,19)(14,24,25,37)(15,29,31,30)(16,28,33,35)(17,34,27,36)(21,26,39,40)(22,38,41,32)(23,42,44,43)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380530);