Finite Group Databasebeta

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Cayley diagram

C2×#8072 #181662

order512
nilpotency class3
isoclinismisoclinic class of #1504
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#8072

Subgroups

subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×#95
prime radicalC25C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

2046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,14,17,15,18,22,25)(13,19,21,26,20,16,23,24);(4,8,10,9)(11,13)(12,16,18,19)(14,21)(15,20)(17,24,25,26)(22,23);(3,5,7,6)(4,9,10,8)(13,20)(16,19)(21,23)(24,26)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239078, #72239084, #72239086, …
maximal normal subgroup of#72239078, #72239084, #72239086, …
maximal quotient of#72239118, #72239594

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88858);