#99278

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#382
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 56 points:

(1,2,3,5,4,6,7,8)(9,10)(11,16)(12,21)(13,23)(14,18)(15,19)(17,22)(20,24)(25,26)(27,35,38,31)(28,39,41,54)(29,34)(30,33)(32,40,45,55)(36,49)(37,48,52,44)(42,46)(43,50)(47,56)(51,53);(9,11,13,19,14,20,22,12)(10,15,17,16,18,21,23,24)(25,27,29,37,30,38,42,52)(26,31,34,44,33,35,46,48)(28,36,40,50,41,51,55,56)(32,43,39,53,45,47,54,49);(9,12,22,20,14,19,13,11)(10,16,23,15,18,24,17,21)(25,28,30,41)(26,32)(27,36,38,51)(29,40,42,55)(31,43)(33,45)(34,39)(35,47)(37,50,52,56)(44,53)(46,54)(48,49)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=dg,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fgh,f^c=fgh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6474);