#7614087

order	512
nilpotency class	3
isoclinism	isoclinic class of #7535047
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3549
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	#59816
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2472 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)(4,19)(5,9)(6,10)(7,13)(8,16)(11,21)(12,23)(15,24)(17,27)(18,26)(20,30)(22,29)(25,32)(28,31);(1,3,6,16,7,18,23,17)(2,8,12,14,13,27,10,26)(4,20,5,15,21,31,22,32)(9,28,19,24,29,30,11,25);(1,4,6,5,7,21,23,22)(2,9,12,19,13,29,10,11)(3,15,17,20,18,32,16,31)(8,24,26,28,27,25,14,30);(2,10,13,12)(8,14,27,26)(9,11,29,19)(24,30,25,28);(1,5,6,21,7,22,23,4)(2,11,12,9,13,19,10,29)(3,15,16,31,18,32,17,20)(8,25,14,28,27,24,26,30)

Polycyclic presentation:

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

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, 7521283);