#475393

order	512
nilpotency class	3
isoclinism	isoclinic class of #460311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2500
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17046
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

669 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eh,d^2=eg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 382589);