#119118

order	512
nilpotency class	3
isoclinism	isoclinic class of #119116
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

522 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=eh,g^b=ghi,e^c=ehi,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26314);