#479875

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

652 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)(3,8)(4,6)(5,7)(9,10)(11,21)(12,26,29,16)(13,20)(14,19)(15,24)(17,30,38,28)(18,32)(22,39,27,37)(23,34,40,33)(25,36)(31,35);(1,3,5,4)(2,6,7,8)(9,11,14,25)(10,15,19,35)(12,27,29,22)(13,31,32,24)(16,34,26,33)(17,37,38,39)(18,36,20,21)(23,30,40,28);(1,4,5,3)(2,6,7,8)(9,12,13,28,14,29,32,30)(10,16,20,38,19,26,18,17)(11,22,24,23,25,27,31,40)(15,33,36,39,35,34,21,37);(3,4)(6,8)(9,12)(10,17,19,38)(11,23)(13,30)(14,29)(15,34,35,33)(16,20,26,18)(21,37,36,39)(22,24)(25,40)(27,31)(28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=efgh,c^2=ef,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,g^c=gi,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, 387071);