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C₂²⋊C₄×#117 #513457

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄²

Aliases

C₂²⋊C₄×#117

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6894 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);(2,4);(9,10,11,12)(13,14)(15,18)(16,17)(19,20);(14,17)(18,20);(13,15,16,19)(14,18,17,20)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²⋊C₄×#525, #72280379, #72280520, …
maximal normal subgroup of	C₂²⋊C₄×#525, #72280379, #72280520, …
maximal quotient of	#72305851, #72306196, C₂×C₂²⋊C₄×#117

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420653);