#265078

order	512
nilpotency class	3
isoclinism	isoclinic class of #265019
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

927 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)(11,21,24,35)(12,18,28,34)(13,14,19,15)(17,29,32,23)(22,33,39,37)(25,31,38,40)(26,27,36,30);(1,3,6,7)(2,4,8,5)(9,11,15,29,16,24,14,23)(10,17,19,35,20,32,13,21)(12,25,30,39,28,38,27,22)(18,33,36,40,34,37,26,31);(2,3)(4,5)(7,8)(9,12)(10,18)(11,22)(13,26)(14,27)(15,30)(16,28)(17,31)(19,36)(20,34)(21,37)(23,38)(24,39)(25,29)(32,40)(33,35);(1,4,6,5)(2,3,8,7)(9,13,14,20,16,19,15,10)(11,21,23,32,24,35,29,17)(12,26,27,34,28,36,30,18)(22,37,38,40,39,33,25,31)

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,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

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, 172274);