#497999

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

600 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)(7,10)(8,12)(9,11)(13,14)(15,23)(16,28)(17,20)(18,32)(19,25)(21,30)(22,29)(24,37)(26,40)(27,44)(31,39)(33,38)(34,42)(35,41)(36,43);(5,7,9,8)(6,10,11,12)(13,15,17,26)(14,19,20,34)(16,24,31,41)(18,33,30,27)(21,37,32,35)(22,38,36,44)(23,39,40,28)(25,43,42,29);(5,8,9,7)(6,10,11,12)(14,20)(19,34)(21,32)(22,36)(23,40)(28,39)(35,37)(38,44);(14,21)(15,24)(16,29)(18,30)(19,23)(20,32)(22,36)(25,27)(26,41)(31,43)(33,42)(34,40)(35,44)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=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, 405195);