#419036

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2222 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326232);