#285854

order	512
nilpotency class	3
isoclinism	isoclinic class of #281659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

610 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)(3,6)(4,5)(7,8)(9,10,14,16,13,17,12,18)(11,19,23,20,22,15,21,24)(25,26)(27,31)(28,33)(29,32)(30,34)(35,40)(36,38)(37,39);(2,5)(6,8)(9,11,12,21,13,22,14,23)(10,15,18,20,17,19,16,24)(25,27,29,37)(26,31,32,39)(28,36,30,35)(33,38,34,40);(1,3,4,7)(2,6,5,8)(9,10,13,17)(11,20,22,24)(12,16,14,18)(15,23,19,21)(26,32)(27,35)(28,30)(31,38)(36,37)(39,40);(2,5)(6,8)(9,12,13,14)(10,16,17,18)(11,21,22,23)(15,24,19,20)(25,28,29,30)(26,33,32,34)(27,36,37,35)(31,38,39,40)

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=e,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 193050);