#6357030

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356901
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#61977	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1579 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264226);