#475184

order	512
nilpotency class	3
isoclinism	isoclinic class of #460311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2500
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17029
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

691 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)(3,14)(4,19,23,10)(5,12)(6,11)(7,13)(8,16)(9,17)(15,27,31,32)(18,25)(20,26,30,24)(21,28,22,29);(1,3,6,16,7,18,5,17)(2,8,11,25,13,9,12,14)(4,15,21,30,23,31,22,20)(10,24,29,32,19,26,28,27);(1,3)(2,9)(4,20)(5,17)(6,16)(7,18)(8,13)(10,27)(11,14)(12,25)(15,21)(19,32)(22,31)(23,30)(24,29)(26,28);(1,4)(2,10,13,19)(3,15)(5,22)(6,21)(7,23)(8,24,9,26)(11,29,12,28)(14,27,25,32)(16,30)(17,20)(18,31)

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 382380);