Finite Group Databasebeta

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Cayley diagram

C2×#31601 #7505152

order512
nilpotency class3
isoclinismisoclinic class of #30412
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#31601

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2052 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);(3,4,7,9)(5,8,6,10)(11,12)(13,15)(14,16)(17,18)(20,23)(24,25);(3,4)(5,8)(6,10)(7,9)(11,13,14,17)(12,15,16,18)(19,20,21,23)(22,24,26,25);(3,5,7,6)(4,8,9,10)(12,16)(15,18)(20,24)(22,26)(23,25);(3,6,7,5)(4,8,9,10)(12,16)(15,18)(20,23)(24,25)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6695188, C2×#6718124, C2×#7505487
maximal normal subgroup ofC2×#6695188, C2×#6718124, C2×#7505487

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7412348);