#467418

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

Polycyclic presentation:

2^9|a^2=efg,b^2=g,c^2=fi,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, 374614);