#199265

order	512
nilpotency class	3
isoclinism	isoclinic class of #9303
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	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1364 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)(11,20,24,31)(12,25)(13,18)(14,28)(15,19)(16,23,32,22)(17,26)(21,36,39,30)(27,33)(29,34)(35,38,40,37);(2,6,8,7)(9,11,13,22,15,24,14,23)(10,16,18,31,19,32,28,20)(12,21,26,37,27,39,29,38)(17,30,33,40,34,36,25,35);(9,12)(10,17)(11,21)(13,26)(14,29)(15,27)(16,30)(18,33)(19,34)(20,35)(22,37)(23,38)(24,39)(25,28)(31,40)(32,36);(1,3,5,4)(2,7,8,6)(9,13,15,14)(10,18,19,28)(11,22,24,23)(12,26,27,29)(16,31,32,20)(17,33,34,25)(21,37,39,38)(30,40,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106461);