#403856

order	512
nilpotency class	3
isoclinism	isoclinic class of #394756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2436
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14883
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1119 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:

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

Polycyclic presentation:

2^9|a^2=fg,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, 311052);