#512256

order	512
nilpotency class	4
isoclinism	isoclinic class of #511896
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=efh,c^2=e,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gh,h^a=hi,c^b=ci,g^c=gi,f^d=fi,g^d=gh,h^d=hi

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, 419452);