Finite Group Databasebeta

← Prev (#181727) Next (#181729) →

C₂×#8098 #181728

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#95
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 18 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214103, #72214132, C₂×#120471, …
maximal normal subgroup of	#72214103, #72214132, C₂×#120471, …
maximal quotient of	C₂×#96443, #72239118

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88924);