#126807

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(6,10)(7,8)(11,12)(13,14,16,21,18,23,30,37,19,24,17,22,29,36,31,38)(15,20,26,33,28,35,40,43,25,32,27,34,39,42,41,44);(2,4)(5,6,8,12,9,11,7,10)(13,15,17,27,18,28,31,41,19,25,16,26,29,39,30,40)(14,20,22,34,23,35,38,44,24,32,21,33,36,42,37,43);(6,11)(10,12)(15,25)(20,32)(26,27)(28,39)(33,34)(35,42)(40,41)(43,44)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fg,c^b=cgi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34003);