#197275

order	512
nilpotency class	3
isoclinism	isoclinic class of #197011
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60415	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1294 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=h,d^2=hi,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104471);