#7519059

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519056
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#58997
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2502 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)(3,17)(4,22)(5,26)(6,30)(7,15)(8,14)(9,16)(10,20)(11,25)(12,29)(13,19)(18,31)(21,32)(23,27)(24,28);(1,3,8,28,9,21,7,20)(2,10,15,32,16,24,14,17)(4,23,26,30,12,13,25,18)(5,27,22,31,11,19,29,6);(1,4)(2,11)(3,18)(5,16)(6,24)(7,25)(8,26)(9,12)(10,31)(13,20)(14,22)(15,29)(17,27)(19,32)(21,30)(23,28);(1,5)(2,12)(3,19)(4,16)(6,28)(7,29)(8,22)(9,11)(10,13)(14,26)(15,25)(17,30)(18,32)(20,31)(21,27)(23,24);(1,6,8,27,9,31,7,19)(2,13,15,18,16,23,14,30)(3,5,28,22,21,11,20,29)(4,24,26,17,12,10,25,32)

Polycyclic presentation:

2^9|b^2=fi,e^2=fi,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,g^c=gi,h^c=hi,g^d=gi

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, 7426255);