#199669

order	512
nilpotency class	3
isoclinism	isoclinic class of #199534
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	#60962	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1126 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fhi,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, 106865);