#195511

order	512
nilpotency class	3
isoclinism	isoclinic class of #18562
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1049 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)(3,11,13,15)(4,14,7,8)(5,10)(6,9)(12,16)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(1,3,5,14,6,13,16,8)(2,7,10,15,9,4,12,11)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(1,4,6,7)(2,8,9,14)(3,12,13,10)(5,11,16,15)(17,19,21,26)(18,22,23,30)(20,28,27,25)(24,32,31,29);(2,9)(3,13)(8,14)(10,12)(18,23)(22,30)(24,31)(29,32)

Polycyclic presentation:

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

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, 102707);