#197370

order	512
nilpotency class	3
isoclinism	isoclinic class of #197321
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1174 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,g^2=h,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, 104566);