#403993

order	512
nilpotency class	3
isoclinism	isoclinic class of #393982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2688 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)(11,21)(12,18)(13,19)(14,17)(15,20)(16,23)(22,36)(24,31)(25,29)(26,32)(27,34)(28,33)(30,37)(35,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,24)(14,23)(15,25)(17,29)(18,30)(19,21)(20,31)(26,39)(27,38)(28,37)(32,36)(33,35)(34,40);(2,5)(6,8)(10,17)(13,14)(16,29)(18,32)(19,20)(21,31)(23,24)(26,28)(30,36)(33,34)(35,40)(37,39);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,22)(13,28)(14,26)(15,27)(16,30)(17,32)(19,33)(20,34)(21,35)(23,39)(24,37)(25,38)(29,36)(31,40)

Polycyclic presentation:

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

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, 311189);