#504034

order	512
nilpotency class	4
isoclinism	isoclinic class of #503866
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂²×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

397 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=f,d^2=ei,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=gh,h^c=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, 411230);