#6361977

order	512
nilpotency class	3
isoclinism	isoclinic class of #6361928
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#61407	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1555 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)(3,9)(4,14)(5,7)(6,12)(8,15)(10,11)(13,16)(17,18)(19,23)(20,27)(21,30)(22,25)(24,26)(28,32)(29,31);(1,3,5,13)(2,6,7,10)(4,11,15,12)(8,16,14,9)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31);(1,2)(3,10)(4,8)(5,7)(6,13)(9,11)(12,16)(14,15)(17,20,22,29)(18,24,25,32)(19,26,21,28)(23,31,30,27);(3,11)(4,15)(6,16)(8,14)(9,10)(12,13)(18,25)(23,30)(24,32)(27,31);(2,7)(6,10)(8,14)(9,16)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269173);