#476661

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:

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

Polycyclic presentation:

2^9|a^2=ef,b^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, 383857);