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Cayley diagram

C22×#2794 #7633561

order512
nilpotency class2
isoclinismisoclinic class of #2668
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC22×#2794

Subgroups

subgroupG/H
center, socle, prime radicalC25C24
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C23 ▷ C1

6002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 6 generators on 28 points:

(1,2);(3,4);(5,6)(7,11)(8,9)(10,12)(14,17)(15,19)(21,22,25,27)(23,24,26,28);(6,9)(10,11)(13,14,16,17)(15,20,19,18)(21,23)(22,24)(25,26)(27,28);(5,7,8,12)(6,10,9,11)(21,24)(22,26)(23,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,15,16,19)(14,18,17,20)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540757);