#265206

order	512
nilpotency class	3
isoclinism	isoclinic class of #265163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18772	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

909 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:

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

Polycyclic presentation:

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

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, 172402);