#510638

order	512
nilpotency class	4
isoclinism	isoclinic class of #510638
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31796
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24806	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2930 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

442 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)(3,8,14,24)(4,12)(5,11)(6,10)(7,22)(9,20)(13,16,29,30)(15,28,17,18)(19,21)(23,32,31,25)(26,27);(1,3,12,17,5,14,6,15)(2,8,4,18,11,24,10,28)(7,23,26,30,20,31,21,16)(9,25,19,29,22,32,27,13);(1,4,12,2,5,10,6,11)(3,13,15,25,14,29,17,32)(7,9,26,27,20,22,21,19)(8,23,18,16,24,31,28,30);(3,14)(7,20)(9,22)(13,29)(15,17)(19,27)(21,26)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417834);