#7567150

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1727 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,25,27,23)(20,28,30,24)(26,31,29,32);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(18,22)(19,25)(20,30)(23,27)(26,31)(29,32);(2,6)(7,8)(10,14)(13,15)(20,30)(24,28)(26,29)(31,32)

Polycyclic presentation:

2^9|a^2=g,c^2=fgi,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7474346);