#194434

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60951	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=h,f^2=hi,b^a=be,c^a=cf,f^a=fh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,g^c=gi

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, 101630);