#403040

order	512
nilpotency class	3
isoclinism	isoclinic class of #396106
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2459
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14865
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1202 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,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, 310236);