#505268

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

442 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)(3,6,7,8)(9,10,13,18)(11,21,24,16)(12,26,28,17)(14,30,27,19)(15,20,29,36)(22,33,31,37)(23,34,40,39)(25,38,32,35);(2,5)(6,8)(9,11,15,32,13,24,29,25)(10,16,20,38,18,21,36,35)(12,23,27,22,28,40,14,31)(17,34,30,33,26,39,19,37);(1,3,4,7)(2,6,5,8)(11,22)(12,27)(14,28)(15,29)(16,33)(17,30)(19,26)(20,36)(21,37)(23,32)(24,31)(25,40)(34,38)(35,39);(2,5)(6,8)(9,12,15,27,13,28,29,14)(10,17,20,30,18,26,36,19)(11,23,32,22,24,40,25,31)(16,34,38,33,21,39,35,37)

Polycyclic presentation:

2^9|a^2=e,b^2=fh,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, 412464);