#418917

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
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

2227 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)(5,6,7,8)(9,10,15,19,14,18,31,36)(11,20,22,30,17,13,23,29)(12,24,28,35,27,37,32,16)(21,33,39,26,34,40,25,38);(1,2,3,4)(9,11)(10,13)(12,25)(14,17)(15,22)(16,33)(18,20)(19,29)(21,28)(23,31)(24,26)(27,39)(30,36)(32,34)(35,40)(37,38);(6,8)(9,12,14,27)(10,16,18,35)(11,21,17,34)(13,26,20,38)(15,32,31,28)(19,37,36,24)(22,25,23,39)(29,33,30,40);(2,4)(9,13)(10,17)(11,18)(12,26)(14,20)(15,30)(16,34)(19,22)(21,35)(23,36)(24,39)(25,37)(27,38)(28,33)(29,31)(32,40)

Polycyclic presentation:

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

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, 326113);