#478284

order	512
nilpotency class	3
isoclinism	isoclinic class of #476293
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

992 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)(3,5)(4,6)(7,8)(9,10)(11,21)(12,26,29,36)(13,20)(14,17)(15,19)(16,22)(18,30,34,28)(23,37,39,32)(24,31)(25,33)(27,35,38,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,27)(13,25)(14,24)(15,22)(17,33)(18,35)(19,31)(20,21)(23,30)(26,37)(28,39)(29,38)(32,36)(34,40);(2,6)(5,8)(10,17)(11,22)(12,28)(13,15)(16,31)(18,36)(21,33)(24,25)(26,34)(27,38)(29,30)(32,37);(3,7)(5,8)(9,12)(10,18)(11,23)(13,30)(14,29)(15,28)(16,32)(17,34)(19,36)(20,26)(21,35)(22,27)(24,39)(25,38)(31,40)(33,37)

Polycyclic presentation:

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

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, 385480);