#197521

order	512
nilpotency class	3
isoclinism	isoclinic class of #197321
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1333 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:

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

Polycyclic presentation:

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

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, 104717);