#494870

order	512
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#459
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=i,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,f^c=fhi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402066);