#460144

order	512
nilpotency class	3
isoclinism	isoclinic class of #453999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

829 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367340);