#456032

order	512
nilpotency class	3
isoclinism	isoclinic class of #454109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16273
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

923 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,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, 363228);