#110094

order	512
nilpotency class	3
isoclinism	isoclinic class of #110086
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator, prime radical	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

586 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=eh,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, 17290);