#505485

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

429 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=ei,d^2=efhi,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, 412681);