#119323

order	512
nilpotency class	3
isoclinism	isoclinic class of #119322
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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=h,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,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, 26519);