#273521

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	#3506	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

841 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=h,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, 180717);