#112047

order	512
nilpotency class	3
isoclinism	isoclinic class of #112035
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

942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,e^c=eh,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, 19243);