#501593

order	512
nilpotency class	4
isoclinism	isoclinic class of #501593
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24708
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

484 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:

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

Polycyclic presentation:

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

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, 408789);