#476709

order	512
nilpotency class	3
isoclinism	isoclinic class of #476707
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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=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, 383905);