#510456

order	512
nilpotency class	4
isoclinism	isoclinic class of #510456
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31613
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3060	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

393 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=eh,d^2=eh,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=def,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,f^d=fi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417652);