#205729

order	512
nilpotency class	3
isoclinism	isoclinic class of #205569
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

996 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)(3,10)(4,9)(5,14)(6,8)(7,11)(12,16)(13,15)(17,18,22,26,23,25,21,19)(20,28,29,32,31,27,30,24);(1,3,5,11,6,12,4,13)(2,7,9,10,8,15,14,16)(17,19,22,25,23,26,21,18)(20,27,29,32,31,28,30,24);(2,8)(7,15)(9,14)(10,16)(17,20,22,29,23,31,21,30)(18,24,19,27,25,32,26,28);(1,4,6,5)(2,9,8,14)(3,11,12,13)(7,16,15,10)(17,21,23,22)(18,19,25,26)(20,29,31,30)(24,28,32,27)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=gi,d^c=di

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, 112925);