#243563

order	512
nilpotency class	3
isoclinism	isoclinic class of #233301
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150759);