#413871

order	512
nilpotency class	3
isoclinism	isoclinic class of #411585
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1365 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=gh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^c=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, 321067);