Finite Group Databasebeta

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Cayley diagram

C2×#24492 #6451904

order512
nilpotency class3
isoclinismisoclinic class of #23860
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#24492

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2433C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6578524, C2×#6454224, C2×#6576756
maximal normal subgroup ofC2×#6578524, C2×#6454224, C2×#6576756

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6359100);