#120726

order	512
nilpotency class	3
isoclinism	isoclinic class of #11247
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#387
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

514 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27922);