Finite Group Databasebeta

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Cayley diagram

C2×#30974 #7502430

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

Aliases

direct productC2×#30974

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2210 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7502671, C2×#7519344, C2×#7515829
maximal normal subgroup ofC2×#7502671, C2×#7519344, C2×#7515829

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409626);