#113347

order	512
nilpotency class	3
isoclinism	isoclinic class of #113347
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	#60958	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

921 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)(3,5)(4,7)(6,8)(9,10)(11,19,21,15)(12,23,26,16)(13,27)(14,17)(18,30)(20,37)(22,32,28,35)(24,33)(25,34,29,36)(31,38)(39,40);(2,5)(3,8)(6,7)(9,11)(10,15)(12,24)(13,28)(14,21)(16,33)(17,19)(18,35)(20,25)(22,30)(23,40)(26,39)(27,32)(29,38)(31,34)(36,37);(1,2)(3,6)(4,7)(5,8)(9,12)(10,16)(11,20)(13,29)(14,26)(15,31)(17,23)(18,36)(19,37)(21,38)(22,39)(24,28)(25,30)(27,34)(32,40)(33,35)

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,f^c=fh,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, 20543);