#122067

order	512
nilpotency class	3
isoclinism	isoclinic class of #121999
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

768 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29263);