#112122

order	512
nilpotency class	3
isoclinism	isoclinic class of #112109
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

840 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=di,f^c=fh,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, 19318);