#4397297

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=i,d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

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, 4304493);