#6624416

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1643 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)(3,8)(4,6)(5,7)(18,23)(19,26)(20,21)(25,28)(27,29)(30,32);(1,3,5,4)(2,6,7,8)(10,14)(12,15)(17,18)(19,24)(20,23)(21,28)(22,25)(26,30)(27,31)(29,32);(9,10)(11,12)(13,14)(15,16)(17,19)(18,24)(20,26)(21,29)(22,27)(23,30)(25,31)(28,32);(1,4,5,3)(2,6,7,8)(9,11,13,16)(10,15,14,12)(18,25)(23,28)(24,31)(30,32);(9,12,13,15)(10,11,14,16)(17,20,22,21)(18,23,25,28)(19,26,27,29)(24,30,31,32)

Polycyclic presentation:

2^9|b^2=fgi,d^2=fghi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282714, #72283140, #72283244
maximal normal subgroup of	#72282714, #72283140, #72283244

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6531612);