#483089

order	512
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

895 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=hi,f^2=ghi,h^2=i,b^a=be,d^a=di,e^a=eg,e^b=eh,f^b=fgh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390285);