#421066

order	512
nilpotency class	3
isoclinism	isoclinic class of #417859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60366	C₂
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)(3,5)(4,6)(7,8)(9,10)(11,21)(12,17)(13,28)(14,31)(15,20)(16,23)(18,30)(19,29)(22,37)(24,40)(25,35)(26,36)(27,34)(32,39)(33,38);(1,3,4,7)(2,5,6,8)(9,11,14,23,15,13,29,30)(10,16,19,21,20,18,31,28)(12,22,25,38,26,24,27,39)(17,33,34,37,36,32,35,40);(2,6)(5,8)(9,12)(10,17)(11,22)(13,24)(14,27)(15,26)(16,32)(18,33)(19,35)(20,36)(21,37)(23,39)(25,29)(28,40)(30,38)(31,34);(9,13)(10,18)(11,15)(12,24)(14,23)(16,20)(17,33)(19,21)(22,26)(25,38)(27,39)(28,31)(29,30)(32,36)(34,40)(35,37)

Polycyclic presentation:

2^9|b^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,e^d=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328262);