#110141

order	512
nilpotency class	3
isoclinism	isoclinic class of #9415
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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:

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

Polycyclic presentation:

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

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, 17337);