Finite Group Databasebeta

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Cayley diagram

C2×#18532 #6348077

order512
nilpotency class3
isoclinismisoclinic class of #18524
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#18532

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC2×#3481C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

3372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72310310, #72310312, C2×#6348971
maximal normal subgroup of#72310310, #72310312, C2×#6348971

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255273);