#4387807

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 2097152
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

1454 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=i,d^2=g,e^2=f,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, 4295003);