#356908

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14,18,16,19,22,17,21)(15,23,25,20,26,27,24,28);(13,15,17,25,19,26,18,24)(14,20,21,27,22,28,16,23);(1,2,4,3)(6,9)(10,12);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16,18,22,19,21,17,14)(15,20,25,27,26,28,24,23)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=ei,g^b=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264104);