#119201

order	512
nilpotency class	3
isoclinism	isoclinic class of #119200
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

520 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)(3,10,13,11)(4,5)(6,9)(7,12,16,14)(8,15)(17,18)(19,27,30,23)(20,24,32,40)(21,26)(22,25)(28,45)(29,43,34,38)(31,48)(33,41,35,44)(36,42)(37,46)(39,47);(1,3)(2,7)(4,11)(5,14)(6,13)(8,10)(9,16)(12,15)(17,19,22,30)(18,23,25,27)(20,31,32,47)(21,34,36,29)(24,39,40,48)(26,43,42,38)(28,35,46,33)(37,44,45,41);(1,4,5,9,6,8,15,2)(3,11,12,7,13,10,14,16)(17,20)(18,24,25,40)(19,28)(21,35)(22,32)(23,37,27,45)(26,44,42,41)(29,47)(30,46)(31,34)(33,36)(38,48,43,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,e^c=ehi,f^c=fhi,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, 26397);