#498285

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3035	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

631 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,f^c=fi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405481);