#501062

order	512
nilpotency class	4
isoclinism	isoclinic class of #501062
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24378
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1046 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408258);