Finite Group Databasebeta

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Cayley diagram

C2×#20778 #6371074

order512
nilpotency class2
isoclinismisoclinic class of #2846
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC2×#20778

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239974, #72239977, #72239978, …
maximal normal subgroup of#72239974, #72239977, #72239978, …
maximal quotient of#72239996

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278270);