#6701464

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1540 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,15,16,14)(17,18)(19,26)(20,30)(21,25)(22,24)(23,29)(27,28)(31,32);(10,13)(11,16)(17,19)(18,23)(20,28)(21,29)(22,31)(24,27)(25,26)(30,32);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(17,18)(19,27)(20,24)(21,25)(22,30)(23,31)(26,28)(29,32);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,21,22)(18,24,25,30)(19,28,29,31)(23,27,26,32);(19,28)(20,22)(23,32)(24,30)(26,27)(29,31)

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6608660);