#434061

order	512
nilpotency class	3
isoclinism	isoclinic class of #434061
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15496
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1175 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341257);