#431637

order	512
nilpotency class	3
isoclinism	isoclinic class of #418470
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15442
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

938 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=f,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 338833);