#498299

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
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

625 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)(5,6,12,17,11,14,30,33,10,15,19,20,9,16,29,34)(7,18,23,35,21,25,13,31,22,8,24,26,28,27,32,36);(2,3)(5,7,19,24,11,21,29,32,10,22,12,23,9,28,30,13)(6,8,20,35,14,27,34,31,15,18,17,26,16,25,33,36);(5,8)(6,13)(7,20)(9,25)(10,18)(11,27)(12,26)(14,24)(15,32)(16,23)(17,22)(19,35)(21,34)(28,33)(29,31)(30,36);(1,3,4,2)(5,9,10,11)(6,14,15,16)(7,21,22,28)(8,25,18,27)(12,29,19,30)(13,24,32,23)(17,33,20,34)(26,31,35,36)

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,d^a=di,f^a=fi,d^b=de,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405495);