#479734

order	512
nilpotency class	3
isoclinism	isoclinic class of #476659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

731 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)(3,5)(4,6)(7,8)(9,10,15,16,14,20,13,19)(11,17,28,32,23,33,21,29)(12,18,26,30,27,34,24,36)(22,31,25,35,38,40,37,39);(1,3,4,7)(2,5,6,8)(10,16)(11,21)(12,24)(13,15)(18,34)(19,20)(22,38)(23,28)(26,27)(29,32)(31,35)(39,40);(9,11,14,23)(10,17,20,33)(12,25,27,37)(13,28,15,21)(16,29,19,32)(18,35,34,39)(22,24,38,26)(30,31,36,40);(2,6)(5,8)(9,12)(10,18)(11,22)(13,24)(14,27)(15,26)(16,30)(17,31)(19,36)(20,34)(21,37)(23,38)(25,28)(29,39)(32,35)(33,40)

Polycyclic presentation:

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

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, 386930);