#6379166

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

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×#578	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4624 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)(5,6,7,8)(9,10,15,19)(11,20,24,22)(12,17,26,34)(13,18,14,28)(16,29,31,23)(21,35,39,36)(25,32,27,33)(30,37,40,38);(9,11)(10,16)(12,21)(13,23)(14,29)(15,24)(17,30)(18,22)(19,31)(20,28)(25,37)(26,39)(27,38)(32,35)(33,36)(34,40);(2,4)(9,12,15,26)(10,17,19,34)(11,21,24,39)(13,27,14,25)(16,30,31,40)(18,33,28,32)(20,35,22,36)(23,38,29,37);(9,10)(11,22)(12,17)(13,18)(14,28)(15,19)(16,29)(20,24)(21,36)(23,31)(25,32)(26,34)(27,33)(30,37)(35,39)(38,40);(9,13,15,14)(10,18,19,28)(11,23,24,29)(12,25,26,27)(16,22,31,20)(17,32,34,33)(21,37,39,38)(30,35,40,36)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286362);