Finite Group Databasebeta

← Prev (#6372175) Next (#6372177) →

C₂²×#1986 #6372176

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

Subgroups

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

4226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#186252, C₂×#110638, #72238871
maximal normal subgroup of	C₂×#186252, C₂×#110638, #72238871
maximal quotient of	C₂×#375492

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279372);