#119366

order	512
nilpotency class	3
isoclinism	isoclinic class of #119362
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×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

581 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:

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

Polycyclic presentation:

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

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, 26562);