#6463584

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442596
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2580
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1693 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)(3,11,14,7)(4,8)(5,10)(6,9)(12,19,22,15)(13,27,29,21)(16,28,30,23)(17,24)(18,25)(20,26)(31,32);(1,3)(2,7)(4,16)(5,19)(6,14)(8,23)(9,11)(10,12)(13,18)(15,20)(17,30)(21,25)(22,26)(24,28)(27,32)(29,31);(1,2)(3,12,14,22)(4,8)(5,10)(6,9)(7,19,11,15)(13,28,29,23)(16,21,30,27)(17,24)(18,25)(20,26)(31,32);(1,4,5,18,6,17,20,31)(2,8,10,25,9,24,26,32)(3,13,15,16,14,29,19,30)(7,21,22,23,11,27,12,28);(4,17)(8,24)(13,29)(16,30)(18,31)(21,27)(23,28)(25,32)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=h,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,g^b=gi,h^b=hi,f^c=fi,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, 6370780);