#449715

order	512
nilpotency class	3
isoclinism	isoclinic class of #417534
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356911);