Finite Group Databasebeta

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Cayley diagram

C2×#11532 #480337

order512
nilpotency class4
isoclinismisoclinic class of #11505
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#11532

Subgroups

subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

1106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#480392, C2×#485442
maximal normal subgroup ofC2×#480392, C2×#485442

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387533);