C₂×#10312 #233354

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

Aliases

direct product

C₂×#10312

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

1776 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)(7,8,12,9)(10,13,11,14)(16,20)(17,21)(18,22);(4,6)(8,13)(9,14)(10,11)(15,16)(17,19)(18,21)(20,22);(7,9,12,8)(10,14,11,13)(15,17,19,16)(18,20,22,21)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#233527, #72216349, #72217013
maximal normal subgroup of	C₂×#233527, #72216349, #72217013
maximal quotient of	C₂×#117091, C₂×#102865

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140550);

C2×#10312 #233354

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#10312 #233354