#6381434

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288630);