#454297

order	512
nilpotency class	3
isoclinism	isoclinic class of #454029
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1277 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:

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

Polycyclic presentation:

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

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, 361493);