#402879

order	512
nilpotency class	3
isoclinism	isoclinic class of #394482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2435
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14860
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1126 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:

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

Polycyclic presentation:

2^9|a^2=eg,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,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, 310075);