#486762

order	512
nilpotency class	4
isoclinism	isoclinic class of #486306
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#2184	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=gh,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393958);