#453813

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

699 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)(3,7)(4,6)(5,8)(9,10)(11,21,24,17)(12,25)(13,16)(14,19)(15,20)(18,28)(22,33,29,30)(23,37,39,40)(26,34)(27,31)(32,38,35,36);(1,3)(2,5)(4,8)(6,7)(10,16,20,19)(11,22,24,29)(12,26,27,28)(17,21)(18,34)(25,31)(30,33)(32,37,35,40);(2,6)(5,7)(9,11,13,22,15,24,14,29)(10,17,16,30,20,21,19,33)(12,23,28,36,27,39,26,38)(18,32,31,40,34,35,25,37);(3,8)(5,7)(9,12)(10,18)(11,23)(13,28)(14,26)(15,27)(16,31)(17,32)(19,25)(20,34)(21,35)(22,36)(24,39)(29,38)(30,40)(33,37)

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 361009);