#197050

order	512
nilpotency class	3
isoclinism	isoclinic class of #9159
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104246);