#479790

order	512
nilpotency class	3
isoclinism	isoclinic class of #477835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2793
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17249
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

623 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)(3,14)(4,17)(5,10,22,28)(6,12)(7,11)(8,13)(9,16)(15,24,31,26)(18,32,20,25)(19,30)(21,29,23,27);(1,3,8,16)(2,9)(4,7,19,6)(5,18,22,20)(10,25)(11,17)(12,30)(13,14)(15,23,31,21)(24,27)(26,29)(28,32);(1,4,6,16,8,19,7,3)(2,9,11,17,13,14,12,30)(5,20,21,31,22,18,23,15)(10,26,27,25,28,24,29,32);(1,5)(2,10)(3,15)(4,18)(6,23)(7,21)(8,22)(9,24)(11,29)(12,27)(13,28)(14,26)(16,31)(17,25)(19,20)(30,32)

Polycyclic presentation:

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

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, 386986);