#403371

order	512
nilpotency class	3
isoclinism	isoclinic class of #394046
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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 310567);