Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(13,14,17,19)(15,23,16,26)(18,28,25,22)(20,27,21,24);(1,2)(3,4)(6,9)(10,11)(14,19)(15,24)(16,27)(18,25)(20,26)(21,23);(13,15,18,24,17,16,25,27)(14,20,22,23,19,21,28,26);(2,4)(13,16,18,27,17,15,25,24)(14,21,22,26,19,20,28,23);(5,7,8,12)(6,10,9,11)(14,19)(15,16)(22,28)(24,27)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307449, #72307649, #72307660, … |
| maximal normal subgroup of | #72307449, #72307649, #72307660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469800);