#511461

order	512
nilpotency class	4
isoclinism	isoclinic class of #511461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32584
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

378 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=ef,f^2=i,g^2=hi,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,e^b=ei,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gh,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418657);