#6674876

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1821 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 6582072);