#479766

order	512
nilpotency class	3
isoclinism	isoclinic class of #476707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

598 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)(3,6)(4,7)(5,8)(9,10,16,20,15,21,14,29)(11,22,26,35,27,38,25,17)(12,13,19,33,30,18,34,32)(23,36,28,39,37,31,40,24);(1,3)(2,5)(4,8)(6,7)(9,11,15,27)(10,17)(12,28)(13,31,18,36)(14,25,16,26)(19,37)(20,22)(21,35)(23,34)(24,32,39,33)(29,38)(30,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,30)(10,18,21,13)(11,23)(14,34,16,19)(17,24)(20,32,29,33)(22,36)(25,40)(26,28)(27,37)(31,38)(35,39);(3,8)(5,6)(9,13,15,18)(10,19,21,34)(11,24,27,39)(12,29,30,20)(14,33,16,32)(17,28,35,40)(22,23,38,37)(25,36,26,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,h^c=hi,e^d=ei,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, 386962);