#6465519

order	512
nilpotency class	3
isoclinism	isoclinic class of #6447057
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2537
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2451 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)(3,13)(4,10)(5,9)(6,11)(7,12)(8,14)(15,26)(16,24)(17,23)(18,31)(19,27)(20,25)(21,28)(22,29)(30,32);(1,3)(2,8)(4,18)(5,16)(6,14)(7,17)(9,13)(10,26)(11,23)(12,24)(15,21)(19,29)(20,30)(22,28)(25,31)(27,32);(2,9,12,11)(3,14,17,16)(10,25,27,28)(15,29,30,18);(1,4,6,21,7,19,5,20)(2,10,11,28,12,27,9,25)(3,15,16,18,17,30,14,29)(8,22,13,26,24,31,23,32);(4,19)(10,27)(15,30)(18,29)(20,21)(22,31)(25,28)(26,32)

Polycyclic presentation:

2^9|c^2=gh,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6372715);