#476615

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
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

1004 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=ei,e^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,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 383811);