#4388244

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 4194304
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

2268 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)(6,8)(9,10,14,15)(11,13,18,16)(12,20,21,17)(19,23,24,22)(28,35)(31,38)(33,39)(36,40);(10,15)(11,18)(12,21)(22,23)(25,26,29,32)(27,30,34,37)(28,31,35,38)(33,36,39,40);(1,2)(3,4)(5,6,7,8)(9,11,14,18)(10,16,15,13)(12,19,21,24)(17,23,20,22)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,2)(3,4)(9,12,14,21)(10,17,15,20)(11,19,18,24)(13,22,16,23)(25,27,29,34)(26,30,32,37)(28,33,35,39)(31,36,38,40);(9,13,14,16)(10,11,15,18)(12,22,21,23)(17,19,20,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fgi,d^2=h,e^2=f,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, 4295440);