Finite Group Databasebeta

← Prev (#193550) Next (#193552) →

C₂×#8951 #193551

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂⁴	C₄×D₄
prime radical	C₂³×D₈	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1808 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);(4,6)(7,8,10,13,11,14,9,12)(15,16,19,21,18,17,20,22);(3,4,5,6)(8,12)(9,10)(13,14)(15,17,19,22,18,16,20,21);(7,9)(10,11)(12,13)(16,17)(21,22)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286602, #72286630, C₂×#208698, …
maximal normal subgroup of	#72286602, #72286630, C₂×#208698, …
maximal quotient of	C₂×#136741, C₂×#136734, C₂×#136737, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100747);