#118390

order	512
nilpotency class	3
isoclinism	isoclinic class of #118389
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25586);