#110713

order	512
nilpotency class	3
isoclinism	isoclinic class of #9431
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

568 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17909);