#394954

order	512
nilpotency class	3
isoclinism	isoclinic class of #394102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23443	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1589 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,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, 302150);