#220916

order	512
nilpotency class	3
isoclinism	isoclinic class of #20286
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#555	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=chi,d^b=de,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128112);