#221323

order	512
nilpotency class	3
isoclinism	isoclinic class of #221164
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61495	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,d^b=de,e^b=eh,f^c=fhi,g^c=gh,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128519);