#198271

order	512
nilpotency class	3
isoclinism	isoclinic class of #198117
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1342 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:

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

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,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,g^c=ghi

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, 105467);