#210408

order	512
nilpotency class	3
isoclinism	isoclinic class of #19072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₄×#120
prime radical	#18984	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

866 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)(3,7)(6,8)(9,10,15,20,16,21,14,35)(11,22,24,17,26,31,25,36)(12,23,28,18,29,30,27,37)(13,19,33,38,34,39,32,40);(3,7)(5,8)(9,11)(10,17)(12,19)(13,30)(14,25)(15,24)(16,26)(18,32)(20,31)(21,36)(22,35)(23,34)(27,40)(28,38)(29,39)(33,37);(1,2)(3,8)(4,6)(5,7)(9,12)(10,18)(11,19)(13,31)(14,27)(15,28)(16,29)(17,32)(20,30)(21,37)(22,34)(23,35)(24,38)(25,40)(26,39)(33,36);(2,6)(5,8)(9,13,15,33,16,34,14,32)(10,19,20,38,21,39,35,40)(11,23,24,18,26,30,25,37)(12,22,28,17,29,31,27,36)

Polycyclic presentation:

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

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, 117604);