#112168

order	512
nilpotency class	3
isoclinism	isoclinic class of #112162
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

838 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:

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

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=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,f^c=fh,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19364);