#4463954

order	512
nilpotency class	2
isoclinism	isoclinic class of #4462490
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2110 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgi,c^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df

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, 4371150);