#108757

order	512
nilpotency class	3
isoclinism	isoclinic class of #108745
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1434
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,11)(4,7)(5,9)(6,10)(8,14)(12,15)(13,16)(17,18)(19,27,29,23)(20,31)(21,26)(22,25)(24,33)(28,36,32,38)(30,41,44,37)(34,45)(35,40)(39,42)(43,47,46,48);(2,7,10,9)(3,12)(8,15,16,11)(13,14)(17,19,21,30,22,29,35,44)(18,23,26,37,25,27,40,41)(20,32,34,46,33,28,42,43)(24,38,39,48,31,36,45,47);(1,3,4,13,6,12,5,14)(2,8,9,15,10,16,7,11)(17,20)(18,24,25,31)(19,28)(21,34)(22,33)(23,36,27,38)(26,39,40,45)(29,32)(30,43)(35,42)(37,47,41,48)(44,46)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,c^b=ch,f^b=fhi,e^c=ehi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15953);