Finite Group Databasebeta

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Cayley diagram

C2×#11875 #482043

order512
nilpotency class4
isoclinismisoclinic class of #11753
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#11875

Subgroups

subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC2×#2535C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

1504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#130752, #72240312, #72240317, …
maximal normal subgroup ofC2×#130752, #72240312, #72240317, …
maximal quotient of#72261216, #72293180, #72293189

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389239);