Finite Group Databasebeta

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Cayley diagram

C2×#22807 #6380632

order512
nilpotency class3
isoclinismisoclinic class of D4×D8
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#22807

Subgroups

subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊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);(3,4,5,6)(7,8)(9,12)(10,14)(11,13)(15,16,18,17,19,21,22,20);(7,9,11,10)(8,12,13,14)(16,20)(17,21)(18,22);(7,10,11,9)(8,12,13,14);(15,17,22,16,19,20,18,21)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D8×#117, C2×#6578130, C2×#6577762
maximal normal subgroup ofC2×D8×#117, C2×#6578130, C2×#6577762

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287828);