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C₄×#2308 #513432

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹⁰×C₄

Aliases

Subgroups

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

8402 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)(7,8)(9,10,11,12)(13,14)(15,19)(16,18)(17,20);(13,15)(14,17)(16,20)(18,19);(6,8)(15,20)(17,19);(14,18)(17,19)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#35346, C₄×#9495, AGL(1,𝔽₅)×#2308, …
maximal normal subgroup of	C₄×#35346, C₄×#9495, #72279892, …
maximal quotient of	C₄×#21776, AGL(1,𝔽₅)×#2308, #72310241, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420628);