#417726

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2424 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,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,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei

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, 324922);