C₂×#7870 #180901

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#7870

Subgroups

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

1960 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);(3,4,6,9,7,10,5,8)(11,12)(13,19)(14,22)(15,18)(16,21)(17,23)(20,25)(24,26);(4,8,10,9)(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

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#180904, #72239955, #72239960, …
maximal normal subgroup of	C₂×#180904, #72239955, #72239960, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88097);

C2×#7870 #180901

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#7870 #180901