#6351371

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#120
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2752 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)(4,13)(5,8)(6,9)(7,10)(12,15)(14,16)(17,18)(19,26)(20,24)(21,29)(22,25)(23,27)(28,31)(30,32);(1,3)(2,4)(5,12)(6,13)(7,14)(8,11)(9,16)(10,15)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(1,4)(2,3)(5,11)(6,16)(7,15)(8,12)(9,13)(10,14)(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,31)(28,32);(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26);(1,5,7,6)(2,8,10,9)(3,12,14,13)(4,11,15,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)

Polycyclic presentation:

2^9|d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209689, #72209690, #72209691
maximal normal subgroup of	#72209689
maximal quotient of	#72209967, #72209972

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258567);