#112343

order	512
nilpotency class	3
isoclinism	isoclinic class of #112331
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×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1159 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:

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

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,e^a=ei,f^a=fh,c^b=cf,d^b=di,e^b=eh,g^b=ghi,e^c=eh,g^c=ghi

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, 19539);