Finite Group Databasebeta

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Cayley diagram

C2×#9479 #201453

order512
nilpotency class3
isoclinismisoclinic class of #9471
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#9479

Subgroups

subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2404C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#249355, C2×#201462, #72284190, …
maximal normal subgroup ofC2×#249355, C2×#201462, #72284190, …
maximal quotient of#72285266, #72285891

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108649);