#498547

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	#3147	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

666 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:

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

Polycyclic presentation:

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

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, 405743);