#273999

order	512
nilpotency class	3
isoclinism	isoclinic class of #273947
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	#61202	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181195);