#411268

order	512
nilpotency class	3
isoclinism	isoclinic class of #401009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 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)(3,8)(4,7)(5,6)(9,10,13,31,15,19,14,20)(11,21,25,35,23,16,24,18)(12,26,30,40,28,17,29,34)(22,32,27,39,38,36,37,33);(2,6)(7,8)(9,11)(10,16,19,21)(12,27)(13,25)(14,24)(15,23)(17,33,26,39)(18,20,35,31)(22,29)(28,37)(30,38)(32,40,36,34);(1,3,5,4)(2,7,6,8)(9,12,13,29,15,28,14,30)(10,17,31,40,19,26,20,34)(11,22,24,27,23,38,25,37)(16,32,35,39,21,36,18,33);(1,4,5,3)(2,7,6,8)(9,11)(10,18,19,35)(12,22)(13,24)(14,25)(15,23)(16,20,21,31)(17,33,26,39)(27,29)(28,38)(30,37)(32,34,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318464);