#498304

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,f^c=fh,h^c=hi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405500);