#395066

order	512
nilpotency class	3
isoclinism	isoclinic class of #394113
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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1512 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)(9,10,13,19)(11,21,24,16)(12,26,27,32)(14,17,15,20)(18,28,33,29)(22,34,25,31)(23,37,38,40)(30,36,35,39);(2,5)(6,8)(9,11)(10,16)(12,23)(13,24)(14,22)(15,25)(17,31)(18,30)(19,21)(20,34)(26,40)(27,38)(28,39)(29,36)(32,37)(33,35);(10,17)(11,22)(14,15)(18,32)(19,20)(23,36)(24,25)(26,33)(28,29)(31,34)(37,40)(38,39);(1,3,4,7)(2,6,5,8)(9,12,13,27)(10,18,19,33)(11,23,24,38)(14,29,15,28)(16,30,21,35)(17,32,20,26)(22,36,25,39)(31,37,34,40)

Polycyclic presentation:

2^9|a^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=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, 302262);