#195524

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1143 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:

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

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=di,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

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102720);