#226443

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(2,6)(7,8)(9,10)(11,18)(12,15)(13,17)(14,21)(16,20)(19,23)(22,24)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40);(1,2,5,6)(3,7,4,8)(11,19)(14,22)(18,23)(21,24)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,3,5,4)(2,7,6,8)(9,11,20,24,13,19,12,21)(10,14,15,23,17,22,16,18)(27,33)(30,36)(34,39)(37,40);(1,4,5,3)(2,7,6,8)(9,12,13,20)(10,15,17,16)(11,21,19,24)(14,23,22,18)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=efh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=chi,d^b=de

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 133639);