#475216

order	512
nilpotency class	3
isoclinism	isoclinic class of #460462
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2499
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17031
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

568 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=e,d^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,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, 382412);