#478389

order	512
nilpotency class	3
isoclinism	isoclinic class of #476443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17145
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,f^c=fi,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, 385585);