#480315

order	512
nilpotency class	4
isoclinism	isoclinic class of #480315
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1464
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #117 ◁ #2873 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

725 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,f^2=h,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,f^b=fg,d^c=di,f^c=fi,f^e=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, 387511);