#498525

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,24)(16,21)(17,30)(18,33)(19,23)(20,27)(22,31)(25,37)(26,38)(28,44)(29,40)(32,34)(35,41)(36,43)(39,42);(1,3,4,2)(6,9)(7,11);(13,15,25,41,18,28,17,27,19,29,31,42,16,26,32,43)(14,20,34,44,21,35,22,24,23,36,30,38,33,39,37,40);(5,7,8,11)(6,10,9,12)(13,16,19,18)(14,21,23,33)(15,26,29,28)(17,25,32,31)(20,35,36,39)(22,30,37,34)(24,38,40,44)(27,41,43,42)

Polycyclic presentation:

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

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, 405721);