#403907

order	512
nilpotency class	3
isoclinism	isoclinic class of #394756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2436
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14884
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

865 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)(3,13,17,27)(4,18)(5,23,16,8)(6,10)(7,12)(9,21)(11,14)(15,30,19,24)(20,32,31,28)(22,25)(26,29);(1,3,14,16,7,17,6,5)(2,8,10,13,12,23,11,27)(4,19,29,20,22,15,21,31)(9,24,25,28,26,30,18,32);(1,4,7,22)(2,9)(3,15,17,19)(5,20,16,31)(6,21,14,29)(8,24)(10,25)(11,18)(12,26)(13,28)(23,30)(27,32);(1,5,6,17,7,16,14,3)(2,8,10,13,12,23,11,27)(4,20,21,19,22,31,29,15)(9,24,25,28,26,30,18,32)

Polycyclic presentation:

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

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, 311103);