#479727

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fi,c^2=ef,d^2=fg,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,d^c=de,h^d=hi

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, 386923);