#119428

order	512
nilpotency class	3
isoclinism	isoclinic class of #119426
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

612 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26624);