#268335

order	512
nilpotency class	3
isoclinism	isoclinic class of #268283
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	#3500	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

920 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fh,g^c=gi,e^d=ei,f^d=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175531);