#474065

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	#16895
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

901 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=gh,c^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,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, 381261);