#395368

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

1422 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,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, 302564);