#467805

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

820 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efg,c^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,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, 375001);