#449348

order	512
nilpotency class	3
isoclinism	isoclinic class of #417460
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1054 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eg,c^2=e,d^2=egi,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi

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, 356544);