Finite Group Databasebeta

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Cayley diagram

C22×#2132 #7622473

order512
nilpotency class4
isoclinismisoclinic class of #353
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC22×#2132

Subgroups

subgroupG/H
centerC22×C4#99
commutatorC2×C4C24×C4
Frattini, cosocleC2×Q8C25
socleC23C2×#99
prime radicalC22×#532C2
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C23×#41 ◁ G
LECSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1

4558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72302925, C2×#493419, #72303481, …
maximal normal subgroup of#72302925, C2×#493419, #72303481, …
maximal quotient of#72303600, #72308672, C2×#480392, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529669);