#450713

order	512
nilpotency class	3
isoclinism	isoclinic class of #417669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15975
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

814 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357909);