#436184

order	512
nilpotency class	3
isoclinism	isoclinic class of #419464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15550
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

945 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=e,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,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, 343380);