#497773

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

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=ehi,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=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, 404969);