#413870

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	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1405 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:

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

Polycyclic presentation:

2^9|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, 321066);