#506261

order	512
nilpotency class	4
isoclinism	isoclinic class of #506261
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#28140
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

574 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)(3,13)(4,19)(5,11)(6,23)(7,12)(8,16)(9,20)(10,22)(14,18)(15,30)(17,27)(21,24)(25,31)(26,32)(28,29);(1,3,6,18,22,32,21,31,7,17,20,28,5,16,4,15)(2,8,9,25,10,13,19,29,12,26,23,30,11,27,24,14);(1,2)(3,14)(4,9)(5,10)(6,24)(7,12)(8,15)(11,22)(13,28)(16,25)(17,29)(18,27)(19,20)(21,23)(26,31)(30,32);(1,4,5,20,7,21,22,6)(2,9,10,19,12,23,11,24)(3,15,16,28,17,31,32,18)(8,25,13,29,26,30,27,14)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413457);