#479757

order	512
nilpotency class	3
isoclinism	isoclinic class of #478457
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

913 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)(3,5,7,8)(9,10)(11,15,21,28)(12,22)(13,17)(14,18)(16,24)(19,26,35,29)(20,31,37,27)(23,30)(25,33)(32,38)(34,39,36,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15,18,28)(12,19,24,35)(13,20)(14,21)(16,29)(17,31,33,27)(22,26)(23,36,38,34)(25,37)(30,40)(32,39);(2,6)(5,8)(9,12)(10,16,18,22)(11,19,21,35)(13,23)(14,24)(15,26)(17,32,33,30)(20,36,37,34)(25,38)(27,39)(28,29)(31,40);(3,7)(5,8)(9,13,14,25)(10,17,18,33)(11,20,21,37)(12,23,24,38)(15,27,28,31)(16,30,22,32)(19,34,35,36)(26,39,29,40)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386953);