#112726

order	512
nilpotency class	3
isoclinism	isoclinic class of #112721
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

1127 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,f^c=fh,g^c=gi

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, 19922);