#268343

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	#25752	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

949 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,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, 175539);