#479867

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17293
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

646 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=ef,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,f^c=fi,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, 387063);