Finite Group Databasebeta

← Prev (#493007) Next (#493009) →

C₄×#2128 #493008

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄³

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×D₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281395, C₄×#12711, C₄×#33553, …
maximal normal subgroup of	#72281395, C₄×#12711, C₄×#33553, …
maximal quotient of	#72283051, #72293113

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400204);