#403459

order	512
nilpotency class	3
isoclinism	isoclinic class of #395190
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2431
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14875
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 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)(3,12)(4,16)(5,19)(6,11)(7,14)(8,15)(9,18)(10,20)(13,23)(17,24)(21,32)(22,28)(25,27)(26,29)(30,31);(1,3)(2,7,11,22)(4,13)(5,14)(6,15)(8,10,12,19)(9,21,16,29)(17,26)(18,27)(20,28)(23,24,25,30)(31,32);(1,3)(2,8,11,12)(4,13)(5,14)(6,15)(7,19,22,10)(9,23,16,25)(17,26)(18,27)(20,28)(21,30,29,24)(31,32);(1,4,5,17,6,18,20,31)(2,9,10,24,11,16,19,30)(3,13,14,26,15,27,28,32)(7,21,12,25,22,29,8,23)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310655);