#4407670

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

1490 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=fg,d^2=gh,e^2=fgi,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, 4314866);