#112846

order	512
nilpotency class	3
isoclinism	isoclinic class of #112845
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

786 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)(19,28,32,24)(20,33)(21,38)(22,27)(23,26)(25,36)(29,46,47,41)(30,40,31,43)(34,39,48,45)(35,44)(37,42);(1,3)(2,6)(4,13)(5,12)(7,8)(9,16)(10,15)(11,14)(17,19,23,32)(18,24,26,28)(20,34,35,48)(21,31,22,30)(25,41,42,46)(27,43,38,40)(29,36,47,37)(33,39,44,45);(1,3)(2,7)(4,13)(5,12)(6,15)(8,16)(9,10)(11,14)(17,20)(18,25)(19,29)(21,37)(22,36)(23,35)(24,39)(26,42)(27,44)(28,45)(30,34)(31,48)(32,47)(33,38)(40,46)(41,43)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dhi,e^c=eh,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, 20042);