#476660

order	512
nilpotency class	3
isoclinism	isoclinic class of #476659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1005 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=i,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi,g^d=gi,h^d=hi

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, 383856);