Finite Group Databasebeta

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Cayley diagram

C2×#24735 #6442796

order512
nilpotency class3
isoclinismisoclinic class of #23848
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#24735

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6444791
maximal normal subgroup ofC2×#6444791

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349992);