#406653

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1294 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=ef,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313849);