#119117

order	512
nilpotency class	3
isoclinism	isoclinic class of #119116
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=eh,g^b=ghi,e^c=ehi,f^c=fi,g^c=gh

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, 26313);