#6461394

order	512
nilpotency class	3
isoclinism	isoclinic class of #6447057
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2572
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1403 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368590);