#115041

order	512
nilpotency class	3
isoclinism	isoclinic class of #115035
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

915 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=dhi,e^c=ehi,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22237);