#273718

order	512
nilpotency class	3
isoclinism	isoclinic class of #273691
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	#62092	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

867 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:

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

Polycyclic presentation:

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

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, 180914);