#4422899

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

2236 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=fg,d^2=g,e^2=fh,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, 4330095);