#205757

order	512
nilpotency class	3
isoclinism	isoclinic class of #205751
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1014 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112953);