C₂×#10315 #233358

order	512
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#10315

Subgroups

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

2120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#233521, #72214375, #72214436, …
maximal normal subgroup of	C₂×#233521, #72214375, #72214436, …
maximal quotient of	C₂×#121387, C₂×#102868, C₂×#117091

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140554);

C2×#10315 #233358

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#10315 #233358