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C₂²×#1483 #6342943

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	Aut(G) order 6291456
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂⁴	C₂×C₂²⋊C₄
prime radical	C₂²×#121	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#18093, C₂²×#17914, C₂²×#17517
maximal normal subgroup of	C₂²×#18093, C₂²×#17914, C₂²×#17517
maximal quotient of	C₂²×#8939

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250139);