#265208

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	#3100	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

890 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)(9,10,15,20)(11,21,25,32)(12,26,29,36)(13,19,14,18)(16,24,33,23)(17,30,35,28)(22,34,27,31)(37,39,40,38);(1,3,6,7)(2,4,8,5)(9,11,14,24,15,25,13,23)(10,16,19,32,20,33,18,21)(12,22,30,39,29,27,28,38)(17,31,36,40,35,34,26,37);(2,3)(4,5)(7,8)(9,12)(10,17)(11,22)(13,28)(14,30)(15,29)(16,31)(18,26)(19,36)(20,35)(21,37)(23,38)(24,39)(25,27)(32,40)(33,34);(1,4,6,5)(2,3,8,7)(9,11,13,23,15,25,14,24)(10,16,18,21,20,33,19,32)(12,27,28,39,29,22,30,38)(17,34,26,40,35,31,36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=ehi,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, 172404);