#273510

order	512
nilpotency class	3
isoclinism	isoclinic class of #273435
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18912	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

923 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=ghi,e^d=ei,f^d=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180706);