#265204

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	#18756	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

912 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)(11,21,24,29)(12,18,28,36)(13,15,19,14)(17,30,33,23)(22,34,39,37)(25,32,38,40)(26,31,35,27);(1,3,6,7)(2,4,8,5)(9,11,14,23,16,24,15,30)(10,17,19,21,20,33,13,29)(12,25,27,22,28,38,31,39)(18,34,35,32,36,37,26,40);(2,3)(4,5)(7,8)(9,12)(10,18)(11,22)(13,26)(14,27)(15,31)(16,28)(17,32)(19,35)(20,36)(21,37)(23,38)(24,39)(25,30)(29,34)(33,40);(1,4,6,5)(2,3,8,7)(9,13,14,10,16,19,15,20)(11,21,23,33,24,29,30,17)(12,26,27,18,28,35,31,36)(22,37,38,40,39,34,25,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,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, 172400);