#453796

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

743 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:

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

Polycyclic presentation:

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

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, 360992);