#6348445

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348364
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

3033 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286361, #72286759, #72286809, …
maximal normal subgroup of	#72286361, #72286759, #72286809, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255641);