#468394

order	512
nilpotency class	3
isoclinism	isoclinic class of #455576
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16609
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

748 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)(3,13)(4,18)(5,10)(6,22)(7,12)(8,16)(9,19)(11,21)(14,25)(15,31)(17,28)(20,26)(23,30)(24,32)(27,29);(1,3)(2,8,12,24)(4,14)(5,15)(6,16)(7,17)(9,25,10,23)(11,13,22,28)(18,31,26,27)(19,29)(20,30)(21,32);(1,4)(2,9,12,10)(3,14)(5,21)(6,19)(7,20)(8,23,24,25)(11,18,22,26)(13,27,28,31)(15,32)(16,29)(17,30);(1,5,6,4,7,19,21,20)(2,10,11,26,12,9,22,18)(3,15,16,14,17,29,32,30)(8,23,13,27,24,25,28,31)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,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, 375590);