#7339798

order	512
nilpotency class	3
isoclinism	isoclinic class of #7339796
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#48088
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=fg,f^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,h^d=hi,f^e=fi,g^e=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7246994);