#479765

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

630 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)(9,10,15,18,14,19,13,30)(11,20,22,33,24,32,23,16)(12,25,28,36,29,17,27,35)(21,34,26,40,37,39,38,31);(1,3)(2,5)(4,8)(6,7)(9,11,14,24)(10,16)(12,26)(13,23,15,22)(17,34,25,39)(18,20)(19,33)(21,27)(28,37)(29,38)(30,32)(31,35,40,36);(1,3)(2,6)(4,8)(5,7)(9,12,14,29)(10,17,19,25)(11,21)(13,27,15,28)(16,31)(18,35,30,36)(20,34)(22,26)(23,38)(24,37)(32,39)(33,40);(3,8)(5,6)(11,22)(12,27)(13,15)(16,32)(17,35)(18,30)(20,33)(23,24)(25,36)(26,38)(28,29)(34,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,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, 386961);