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C₄×#1455 #180836

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂×C₄²×C₈

Aliases

Subgroups

	subgroup	G/H
center	C₄³	C₂³
commutator	C₂²	C₂×C₄³
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₄²
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#1455, C₄×#19745, C₄×#11669
maximal normal subgroup of	C₄×#19745, C₄×#11669
maximal quotient of	AGL(1,𝔽₅)×#1455, C₄×#7846

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88032);