#4419462

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2185 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=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, 4326658);