#197233

order	512
nilpotency class	3
isoclinism	isoclinic class of #196993
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1269 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104429);