#6381448

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
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

3185 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)(11,17)(12,18)(13,19)(14,16)(15,20)(21,30)(22,28)(23,32)(24,34)(25,33)(26,31)(27,29)(35,40)(36,38)(37,39);(5,7,8,6)(10,16)(12,24)(13,14)(17,28)(18,33)(19,20)(21,35)(22,26)(25,27)(30,39)(31,32)(36,37);(2,4)(9,11,15,23)(10,17,20,32)(12,21,25,37)(13,26,14,22)(16,28,19,31)(18,30,33,39)(24,35,27,36)(29,38,34,40);(9,12)(10,18)(11,21)(13,24)(14,27)(15,25)(16,29)(17,30)(19,34)(20,33)(22,36)(23,37)(26,35)(28,38)(31,40)(32,39);(9,13,15,14)(10,19,20,16)(11,22,23,26)(12,24,25,27)(17,28,32,31)(18,34,33,29)(21,36,37,35)(30,38,39,40)

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,f^a=fi,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288644);