#411269

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

Polycyclic presentation:

2^9|a^2=efg,b^2=gh,c^2=fgi,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, 318465);