#108751

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×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=i,f^2=h,g^2=h,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, 15947);