Finite Group Databasebeta

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Cayley diagram

C2×#30898 #7502185

order512
nilpotency class3
isoclinismisoclinic class of #30412
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#30898

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2816 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);(3,4)(5,6)(7,8,10,11)(9,13,14,12)(15,16,18,19)(17,20,22,21);(8,11)(9,14)(15,17)(16,20)(18,22)(19,21);(4,6)(16,19)(17,21)(20,22);(4,6)(7,9,10,14)(8,12,11,13)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7502671, C2×#7518310, C2×#7515320
maximal normal subgroup ofC2×#7502671, C2×#7518310, C2×#7515320

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409381);