Finite Group Databasebeta

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Cayley diagram

C2×#12591 #492913

order512
nilpotency class2
isoclinismisoclinic class of #120
automorphism?
homologyAbelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#12591

Subgroups

subgroupG/H
centerC23×C8C23
commutatorC22C22×C4×C8
Frattini, cosocleC22×C8C24
socle, prime radicalC24C22×C8
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 4 generators on 30 points:

(1,2);(7,8)(9,12)(10,11)(13,14)(15,16,17,21,18,22,23,28,19,20,24,26,25,27,29,30);(3,4,5,6)(8,11)(12,14);(7,9,10,13)(8,12,11,14)(16,20)(21,26)(22,27)(28,30)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#93250
maximal normal subgroup ofC2×#93250
maximal quotient ofC2×#128204, C2×#128226, C2×#151881

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400109);