#471374

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

542 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)(9,10,15,30,16,19,14,13)(11,20,31,17,24,22,23,34)(12,25,28,18,29,35,32,27)(21,33,37,39,26,40,38,36);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17)(12,26,29,21)(13,22)(14,31)(15,23)(16,24)(18,33,27,40)(19,34)(20,30)(25,39,35,36)(28,37,32,38);(3,7)(5,8)(9,12,15,28,16,29,14,32)(10,18,13,25,19,27,30,35)(11,21,23,38,24,26,31,37)(17,33,22,39,34,40,20,36);(9,13,15,19,16,30,14,10)(11,22,23,34,24,20,31,17)(12,27,28,35,29,18,32,25)(21,33,38,39,26,40,37,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,c^2=f,d^2=f,e^2=i,f^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,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,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, 378570);