C₂×#7882 #180942

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#7882

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#95
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1926 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#196174, C₂×#196180, C₂×#181445
maximal normal subgroup of	C₂×#196174, C₂×#196180, C₂×#181445
maximal quotient of	#72309247

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88138);

C2×#7882 #180942

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#7882 #180942