Finite Group Databasebeta

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Cayley diagram

C2×#10402 #238429

order512
nilpotency class3
isoclinismisoclinic class of #9914
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

direct productC2×#10402

Subgroups

subgroupG/H
center, socleC24#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#2656C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267808, #72267816, C2×#241438, …
maximal normal subgroup of#72267808, #72267816, C2×#241438, …
maximal quotient of#72267816

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145625);