Finite Group Databasebeta

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Cayley diagram

C2×#20347 #6369749

order512
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphism?
homologyAbelianization (H1) = C23×C42 · Schur multiplier (H2) = C210×C4

Aliases

direct productC2×#20347

Subgroups

subgroupG/H
centerC23×C4C2×D4
commutatorC4C23×C42
Frattini, cosocleC22×C4C25
socleC24C22×D4
prime radicalC26D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

8522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235537, C2×#123828, #72238818, …
maximal normal subgroup of#72235537, C2×#123828, #72238818, …
maximal quotient of#72235607, #72238824, C2×#181727, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276945);