#112724

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

775 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)(17,18,22,27,23,26,21,36)(19,28,32,44,30,24,31,39)(20,25,34,43,35,41,37,42)(29,38,33,48,46,45,47,40);(1,3)(2,6)(4,13)(5,12)(7,8)(9,16)(10,15)(11,14)(17,19)(18,24)(20,33)(21,32)(22,31)(23,30)(25,40)(26,28)(27,44)(29,34)(35,47)(36,39)(37,46)(38,42)(41,48)(43,45);(1,3)(2,7)(4,13)(5,12)(6,15)(8,16)(9,10)(11,14)(17,20,23,35)(18,25,26,41)(19,29,30,46)(21,37,22,34)(24,38,28,45)(27,43,36,42)(31,47,32,33)(39,48,44,40)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,f^2=h,g^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, 19920);