#273561

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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

837 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)(3,7)(4,8)(9,10)(11,19,23,16)(12,14)(13,24,27,17)(15,18)(20,22,32,29)(21,30)(25,28)(26,37,39,31)(33,35)(34,36)(38,40);(1,2,5,3)(4,7,8,6)(9,11,14,23)(10,16,12,19)(13,21,27,35)(15,29,28,22)(17,30,24,33)(18,32,25,20)(26,38,39,36)(31,40,37,34);(9,12,14,10)(11,20,23,32)(13,24,27,17)(15,18,28,25)(16,29,19,22)(21,34,35,40)(26,31,39,37)(30,38,33,36);(1,3,5,2)(4,6,8,7)(9,13)(10,17)(11,21)(12,24)(14,27)(15,26)(16,30)(18,31)(19,33)(20,34)(22,36)(23,35)(25,37)(28,39)(29,38)(32,40)

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=hi,d^2=i,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, 180757);