#417725

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×#454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2392 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)(9,10,15,20)(11,21,24,29)(12,18,26,33)(13,16,14,19)(17,28,32,22)(23,35,38,39)(25,34,27,30)(31,37,40,36);(10,16)(11,22)(12,25)(13,14)(18,33)(19,20)(21,29)(24,28)(26,27)(31,39)(35,40)(36,37);(2,4)(9,11)(10,17)(12,23)(13,22)(14,28)(15,24)(16,29)(18,31)(19,21)(20,32)(25,37)(26,38)(27,36)(30,39)(33,40)(34,35);(6,8)(9,12,15,26)(10,18,20,33)(11,23,24,38)(13,27,14,25)(16,30,19,34)(17,31,32,40)(21,35,29,39)(22,36,28,37)

Polycyclic presentation:

2^9|a^2=i,d^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, 324921);