#471159

order	512
nilpotency class	3
isoclinism	isoclinic class of #461251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

520 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fh,c^2=e,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,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, 378355);