#197163

order	512
nilpotency class	3
isoclinism	isoclinic class of #197011
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	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1227 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=h,g^2=i,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, 104359);