C₄×#2230 #495997

order	512
nilpotency class	4
isoclinism	isoclinic class of #2230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₄×#2230

Subgroups

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#476
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

586 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);(6,9)(7,15,17,12)(8,10)(13,14)(16,18,19,20);(5,6,8,9,11,14,10,13)(7,12,16,18,17,15,19,20);(5,7)(6,12)(8,16)(9,18)(10,19)(11,17)(13,20)(14,15)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#2230, C₄×#33688, C₄×#12364, …
maximal normal subgroup of	C₄×#33688, C₄×#12364, C₄×#12366
maximal quotient of	AGL(1,𝔽₅)×#2230, C₄×#12823, C₄×#13186

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403193);

C4×#2230 #495997

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₄×#2230 #495997