Finite Group Databasebeta

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Cayley diagram

C2×#8455 #186804

order512
nilpotency class3
isoclinismisoclinic class of #8453
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#8455

Subgroups

subgroupG/H
center, socleC23#379
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#3486C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

3574 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)(12,16,18,21)(17,25,26,24);(3,5)(4,7)(6,10)(8,9)(11,12)(13,17)(14,21)(15,18)(16,23)(19,24)(20,26)(22,25);(11,13,15,20)(12,17,18,26)(14,22,23,19)(16,24,21,25)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#144629, #72210075, #72210076, …
maximal normal subgroup ofC2×#144629, #72210075, #72210076, …
maximal quotient of#72238886, #72261973, C22×#8455, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94000);