C₂×#10036 #232746

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#10036

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	#392
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1452 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);(3,4,8,5)(6,9,7,10)(11,12)(13,19)(14,22)(15,18)(16,21)(17,24)(20,26)(23,25);(4,9)(5,10)(6,7)(16,25)(19,26);(3,5,8,4)(6,10,7,9)(11,13)(12,16)(14,23)(15,20)(17,26)(18,25)(19,22)(21,24)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#232766, C₂×#244118, C₂×#244123
maximal normal subgroup of	C₂×#232766, C₂×#244118, C₂×#244123
maximal quotient of	C₂×#121341, C₂×#102868, C₂×#102869, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139942);

C2×#10036 #232746

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#10036 #232746