#395369

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×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1360 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^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, 302565);