Finite Group Databasebeta

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Cayley diagram

C2×#20715 #6370918

order512
nilpotency class2
isoclinismisoclinic class of #2846
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC2×#20715

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
socleC24C23×C4
prime radicalC26C2×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

7354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72237122, C2×#6629283, #72237157, …
maximal normal subgroup of#72237122, C2×#6629283, #72237157, …
maximal quotient of#72239289, #72239312, #72264190, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278114);