#396529

order	512
nilpotency class	3
isoclinism	isoclinic class of #394044
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1874 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=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, 303725);