Finite Group Databasebeta

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Cayley diagram

C2×#18163 #6346417

order512
nilpotency class3
isoclinismisoclinic class of #18162
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#18163

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3490C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

4488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268101, #72268103, C2×#6346594, …
maximal normal subgroup of#72268101, #72268103, C2×#6346594, …
maximal quotient ofC2×#485685, #72285734, #72285796

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253613);