#505340

order	512
nilpotency class	4
isoclinism	isoclinic class of #501868
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1771 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

405 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,c^2=ei,d^2=fh,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,d^c=dh,g^c=ghi,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412536);