#343759

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

477 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:

(1,2)(3,4)(5,7)(6,8)(9,10,14,15,13,18,11,16)(12,20,19,24,22,23,21,17);(1,3)(2,6)(4,7)(5,8)(9,11,13,14)(10,15,18,16)(12,21,22,19)(17,20,24,23)(25,26,30,33)(27,34,36,39)(28,32,29,37)(31,38,40,35);(1,4,5,6)(2,3,7,8)(10,16)(11,14)(15,18)(17,20)(19,21)(23,24)(25,27,30,36)(26,31,33,40)(28,35,29,38)(32,39,37,34);(2,7)(3,8)(9,12,13,22)(10,17,18,24)(11,19,14,21)(15,23,16,20)(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=ehi,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250955);