#493741

order	512
nilpotency class	4
isoclinism	isoclinic class of #480315
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×#1799
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1101 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^c=fi,g^c=ghi,g^f=gi

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, 400937);