#119145

order	512
nilpotency class	3
isoclinism	isoclinic class of #119140
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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

504 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=eh,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, 26341);