#462443

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

912 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:

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

Polycyclic presentation:

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

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, 369639);