#473304

order	512
nilpotency class	3
isoclinism	isoclinic class of #456658
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

703 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:

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

Polycyclic presentation:

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

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, 380500);