#7500843

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

3154 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)(9,10,15,21,16,22,14,20)(11,23,27,29,28,37,26,17)(12,18,30,25,31,32,35,39)(13,24,33,38,34,40,36,19);(5,6)(7,8)(9,11,16,28)(10,17,22,29)(12,24,31,40)(13,32,34,18)(14,27,15,26)(19,35,38,30)(20,23,21,37)(25,36,39,33);(2,4)(6,8)(9,12,16,31)(10,18,22,32)(11,24,28,40)(13,29,34,17)(14,35,15,30)(19,27,38,26)(20,39,21,25)(23,33,37,36);(2,4)(9,13)(10,19)(11,25)(12,29)(14,33)(15,36)(16,34)(17,31)(18,27)(20,24)(21,40)(22,38)(23,30)(26,32)(28,39)(35,37);(9,14,16,15)(10,20,22,21)(11,26,28,27)(12,30,31,35)(13,33,34,36)(17,37,29,23)(18,25,32,39)(19,24,38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408039);