#119164

order	512
nilpotency class	3
isoclinism	isoclinic class of #119158
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,e^c=ehi,f^c=fhi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26360);