#6348115

order	512
nilpotency class	3
isoclinism	isoclinic class of #18524
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3008 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,29)(20,32)(21,27)(22,28)(23,26)(24,30)(25,31);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,24)(20,21)(22,30)(23,31)(25,28)(26,32)(27,29);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,23,30)(18,25,26,29)(19,21,31,22)(24,28,32,27);(2,9)(7,16)(8,13)(10,15)(18,26)(24,32)(25,29)(27,28);(17,21,23,22)(18,27,26,28)(19,20,31,30)(24,29,32,25)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283723
maximal normal subgroup of	#72283723

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255311);