Finite Group Databasebeta

← Prev (#354020) Next (#354022) →

C₂×#10922 #354021

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

Subgroups

	subgroup	G/H
center	C₂⁴×C₄	C₂³
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂⁵	C₂²×C₄
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4182 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)(8,10)(15,16,19,21,18,17,20,22);(4,6)(11,12,13,14)(16,17)(21,22);(7,8,9,10)(15,17,19,22,18,16,20,21)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₄×#1955, #72215970, #72215994, …
maximal normal subgroup of	C₂×C₄×#1955, #72215970, #72215994, …
maximal quotient of	C₂×#110284, C₂×#147372, #72320092, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261217);