#480375

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×#1467
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #2873 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

665 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,e^2=i,f^2=h,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,f^b=fgi,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, 387571);