Finite Group Databasebeta

← Prev (#181518) Next (#181520) →

C₂×#8023 #181519

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁶	D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

3714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#194755, C₂×#192094, C₂×#182644, …
maximal normal subgroup of	C₂×#194755, C₂×#192094, C₂×#182644, …
maximal quotient of	C₂×#127647, C₂×#127645

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88715);