#110246

order	512
nilpotency class	3
isoclinism	isoclinic class of #110242
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

603 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)(3,11,14,8)(4,6)(5,10)(7,9)(12,15,16,13)(17,18)(19,28)(20,25,33,41)(21,35)(22,26)(23,27)(24,30)(29,42,44,48)(31,39)(32,45,34,40)(36,38)(37,46,47,43);(1,3,7,13,5,14,6,15)(2,8,4,12,10,11,9,16)(17,19,22,30,23,31,21,36)(18,24,26,39,27,38,35,28)(20,29,34,43,33,44,32,46)(25,37,40,48,41,47,45,42);(1,4,7,10,5,9,6,2)(3,12,13,11,14,16,15,8)(17,20,23,33)(18,25)(19,29,31,44)(21,32,22,34)(24,37)(26,40)(27,41)(28,42)(30,43,36,46)(35,45)(38,47)(39,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17442);