#431484

order	512
nilpotency class	3
isoclinism	isoclinic class of #427241
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15439
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1086 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=ef,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 338680);