Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| 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×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,21,16,27)(17,24,19,22)(20,25,28,26);(1,2,3,4)(13,15,17,26,18,16,19,25)(14,20,22,21,23,28,24,27);(6,9)(10,12)(13,16,18,15)(14,21,23,27)(17,26,19,25)(20,24,28,22);(13,15,18,16)(14,21,23,27)(17,25,19,26)(20,24,28,22);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,23,24)(15,25,16,26)(20,27,28,21)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72282839, #72282971, #72282981, … |
| maximal normal subgroup of | #72282839, #72282971, #72282981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361968);