#459881

order	512
nilpotency class	3
isoclinism	isoclinic class of #454109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16351
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

929 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)(3,15,17,5)(4,10)(6,12)(7,14)(8,13)(9,11,24,18)(16,23,19,31)(20,29,26,32)(21,27)(22,28)(25,30);(1,3,6,18,8,17,7,11)(2,9,12,5,13,24,14,15)(4,19,21,29,22,16,25,32)(10,26,27,23,28,20,30,31);(1,4,6,21,8,22,7,25)(2,10,12,27,13,28,14,30)(3,16,11,29,17,19,18,32)(5,23,9,26,15,31,24,20);(1,5,6,24,8,15,7,9)(2,11,12,3,13,18,14,17)(4,20,21,31,22,26,25,23)(10,16,27,32,28,19,30,29)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367077);