#112722

order	512
nilpotency class	3
isoclinism	isoclinic class of #112721
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

798 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)(3,10,12,7)(4,8)(5,9)(6,11,16,13)(14,15)(17,18)(19,27,29,23)(20,24)(21,34)(22,25)(26,37)(28,38,46,45)(30,39,35,43)(31,48,47,40)(32,44)(33,41)(36,42);(1,3)(2,6)(4,13)(5,12)(7,8)(9,16)(10,15)(11,14)(17,19)(18,23)(20,31)(21,35)(22,29)(24,40)(25,27)(26,43)(28,32)(30,37)(33,47)(34,39)(36,46)(38,42)(41,48)(44,45);(1,3)(2,7)(4,13)(5,12)(6,15)(8,16)(9,10)(11,14)(17,20,22,33)(18,24,25,41)(19,28,29,46)(21,36,37,32)(23,38,27,45)(26,44,34,42)(30,47,35,31)(39,48,43,40)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,f^c=fh,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, 19918);