#476268

order	512
nilpotency class	3
isoclinism	isoclinic class of #476175
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23399	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1269 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)(4,7)(5,8)(9,10,15,20,16,21,14,19)(11,22,25,17,12,26,24,34)(13,28,31,36,32,18,30,37)(23,38,29,40,27,35,39,33);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,16)(13,23)(14,25)(15,24)(18,33)(19,26)(20,22)(21,34)(27,32)(28,40)(29,30)(31,39)(35,37)(36,38);(1,3)(2,6)(4,8)(5,7)(9,12,16,11)(10,17,21,34)(13,29,32,39)(14,24,15,25)(18,35,28,38)(19,26,20,22)(23,30,27,31)(33,37,40,36);(3,8)(5,6)(9,13)(10,18)(11,23)(12,27)(14,31)(15,30)(16,32)(17,33)(19,37)(20,36)(21,28)(22,38)(24,29)(25,39)(26,35)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383464);