Finite Group Databasebeta

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Cayley diagram

C2×#24077 #6452930

order512
nilpotency class3
isoclinismisoclinic class of #23906
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#24077

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3492C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6453841, #72240575, #72240576, …
maximal normal subgroup ofC2×#6453841, #72240575, #72240576, …
maximal quotient of#72261875, #72264535

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6360126);