#6370815

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=di,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278011);