Finite Group Databasebeta

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Cayley diagram

C2×#25923 #6447716

order512
nilpotency class3
isoclinismisoclinic class of #24464
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#25923

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3262 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);(3,4,7,10)(5,9,6,8)(11,12)(13,17)(14,23)(15,19)(16,20)(18,24)(21,26)(22,25);(3,5,7,6)(4,8,10,9)(11,13)(12,17)(14,24)(15,21)(16,22)(18,23)(19,26)(20,25);(3,5,7,6)(4,9,10,8)(11,14,15,13,16,21,24,22)(12,17,18,23,20,25,19,26);(3,6,7,5)(4,9,10,8)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72271446, #72271563
maximal normal subgroup of#72271446, #72271563

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354912);