#411110

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=gh,c^2=fg,d^2=f,e^2=i,h^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,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318306);