C₂×#8093 #181725

order	512
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#8093

Subgroups

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

1728 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);(4,8)(5,6)(9,10)(11,12,15,18,16,19,14,23)(13,20,22,17,21,26,24,25);(3,4,6,10,7,9,5,8)(11,13)(12,17)(14,24)(15,22)(16,21)(18,26)(19,25)(20,23);(4,9)(8,10)(13,21)(17,25)(20,26)(22,24)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#181727
maximal normal subgroup of	C₂×#181727

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88921);

C2×#8093 #181725

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#8093 #181725