#461123

order	512
nilpotency class	3
isoclinism	isoclinic class of #455830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16380
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

881 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)(3,14)(4,20,23,27)(5,24)(6,12,7,26)(9,18)(10,16,31,22)(11,15)(17,29)(19,28)(21,32)(25,30);(1,3)(2,9,13,29)(4,21)(5,16)(6,17)(7,18)(8,19)(10,30,31,32)(11,27,24,20)(12,14,26,28)(15,22)(23,25);(1,4)(2,10)(3,15)(5,19)(6,22)(7,16)(8,23)(9,11)(12,27)(13,31)(14,30)(17,25)(18,21)(20,26)(24,29)(28,32);(1,5)(2,11)(3,16)(4,17)(6,21)(7,25)(8,15)(9,27)(10,28)(12,30)(13,24)(14,31)(18,23)(19,22)(20,29)(26,32)

Polycyclic presentation:

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

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, 368319);