#6687393

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682638
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=fg,d^2=fi,e^2=fi,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,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594589);