#467417

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
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

762 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)(3,7)(4,5)(6,8)(9,10)(11,21)(12,26,28,33)(13,19)(14,18)(15,20)(16,22)(17,27,34,29)(23,30,38,40)(24,32)(25,31)(35,36,39,37);(2,5)(6,7)(9,11,14,24,15,25,13,22)(10,16,19,31,20,32,18,21)(12,23,27,37,28,38,29,36)(17,30,33,39,34,40,26,35);(1,3,4,8)(2,6,5,7)(11,22)(12,27)(13,14)(16,21)(17,33)(18,19)(24,25)(26,34)(28,29)(31,32)(35,39)(36,37);(3,8)(6,7)(9,12)(10,17)(11,23)(13,29)(14,27)(15,28)(16,30)(18,26)(19,33)(20,34)(21,35)(22,36)(24,37)(25,38)(31,39)(32,40)

Polycyclic presentation:

2^9|a^2=efg,b^2=g,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,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, 374613);