#193818

order	512
nilpotency class	3
isoclinism	isoclinic class of #29610
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1190 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,c^b=ci,e^b=eh,f^b=fh,g^b=gh,e^c=eh,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101014);