#493297

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,f^2=h,g^2=h,h^2=i,b^a=be,c^a=ci,d^a=de,c^b=cei,d^b=di

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, 400493);