Finite Group Databasebeta

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Cayley diagram

C2×#22805 #6380630

order512
nilpotency class3
isoclinismisoclinic class of D4×D8
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#22805

Subgroups

subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2478 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);(3,4,5,6)(7,8)(9,12)(10,14)(11,13)(15,16,19,17,20,22,18,21);(7,9,11,10)(8,12,13,14)(16,17)(18,19)(21,22);(7,10,11,9)(8,12,13,14);(15,17,18,16,20,21,19,22)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6578177, C2×#6577769, C2×#6380924
maximal normal subgroup ofC2×#6578177, C2×#6577769, C2×#6380924

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287826);