#4387655

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1702 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,d^2=g,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, 4294851);