#7502166

order	512
nilpotency class	3
isoclinism	isoclinic class of #7502165
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2675
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2520 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)(3,14)(4,18)(5,23)(6,13)(7,12)(8,11)(9,16)(10,21)(15,25)(17,29)(19,26)(20,30)(22,28)(24,27)(31,32);(1,3,8,17)(2,9,11,23)(4,19,22,20)(5,7,16,6)(10,26,27,30)(12,29,13,14)(15,24,31,21)(18,32,28,25);(1,4)(2,10)(3,15)(5,20)(6,21)(7,24)(8,22)(9,25)(11,27)(12,18)(13,28)(14,30)(16,19)(17,31)(23,32)(26,29);(1,5,6,3,8,16,7,17)(2,9,12,14,11,23,13,29)(4,20,21,15,22,19,24,31)(10,25,18,30,27,32,28,26);(2,11)(9,23)(10,27)(12,13)(14,29)(18,28)(25,32)(26,30)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409362);