#498524

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,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, 405720);