#476482

order	512
nilpotency class	3
isoclinism	isoclinic class of #476297
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1049 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 383678);