#6381426

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
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

3180 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)(11,16)(12,23)(13,15)(14,19)(17,25)(18,26)(20,34)(21,31)(22,28)(24,29)(27,33)(30,36)(32,37)(35,39)(38,40);(5,7,8,6)(10,15)(12,24)(13,26)(16,28)(17,23)(18,19)(20,35)(22,37)(25,27)(30,34)(31,32)(36,38);(2,4)(9,11)(10,16)(12,20)(13,22)(14,21)(15,28)(17,30)(18,32)(19,31)(23,34)(24,35)(25,36)(26,37)(27,38)(29,39)(33,40);(9,12,14,25)(10,17,19,23)(11,20,21,36)(13,27,26,24)(15,29,18,33)(16,30,31,34)(22,38,37,35)(28,39,32,40);(11,21)(16,31)(20,36)(22,37)(28,32)(30,34)(35,38)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^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, 6288622);