#112609

order	512
nilpotency class	3
isoclinism	isoclinic class of #112605
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	#35329	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19805);