#108685

order	512
nilpotency class	3
isoclinism	isoclinic class of #108682
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

419 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:

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

Polycyclic presentation:

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

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, 15881);