#4389468

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=gi,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, 4296664);