#5123778

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2325 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=ghi,c^2=gh,d^2=fhi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5030974);