#505035

order	512
nilpotency class	4
isoclinism	isoclinic class of #505034
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#27110
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1765 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

418 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)(3,6)(4,18)(5,17)(7,13)(8,12)(9,14)(10,22)(11,25)(15,27)(16,26)(19,32)(20,31)(21,29)(23,28)(24,30);(1,3,8,14,9,12,6,2)(4,19,27,28,22,31,21,30)(5,16,7,13,26,17,25,11)(10,23,15,32,18,24,29,20);(1,4,6,21,9,22,8,27)(2,10,12,29,14,18,3,15)(5,24,25,32,26,23,7,20)(11,31,17,28,13,19,16,30);(1,5,6,25,9,26,8,7)(2,11,12,17,14,13,3,16)(4,20,27,23,22,32,21,24)(10,28,15,31,18,30,29,19)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412231);