#449171

order	512
nilpotency class	3
isoclinism	isoclinic class of #421447
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2474
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15934
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

706 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:

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

Polycyclic presentation:

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

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, 356367);