#198929

order	512
nilpotency class	3
isoclinism	isoclinic class of #9279
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60940	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1376 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106125);