#415687

order	512
nilpotency class	3
isoclinism	isoclinic class of #394590
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

849 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=gh,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi

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, 322883);