Finite Group Databasebeta

← Prev (#6443750) Next (#6443752) →

Cayley diagram

C2×#26787 #6443751

order512
nilpotency class3
isoclinismisoclinic class of #23848
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#26787

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#1999C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2738 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

(1,2);(4,5)(7,8)(9,14)(10,12)(11,13)(16,19)(20,22);(3,4,6,5)(7,9,11,10)(8,12,13,14)(16,20)(18,21)(19,22);(3,4,6,5)(7,10,11,9)(8,12,13,14)(15,16,17,19)(18,22,21,20);(3,5,6,4)(8,13)(12,14)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72264655, #72264693, C2×#7613557, …
maximal normal subgroup of#72264655, #72264693, C2×#7613557, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350947);