#342042

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

629 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=hi,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 249238);