#4934219

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
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

2102 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)(5,6)(7,8)(9,10,14,15)(11,13,19,16)(12,20,22,17)(18,23,24,21)(27,33)(29,37)(31,34)(32,36);(10,15)(13,16)(17,20)(21,23)(25,26,30,28)(27,34,33,31)(29,32,37,36)(35,40,39,38);(6,7)(9,11)(10,16)(12,18)(13,15)(14,19)(17,23)(20,21)(22,24)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(1,2)(3,4)(5,7,8,6)(9,12,14,22)(10,17,15,20)(11,18,19,24)(13,21,16,23)(25,28,30,26)(27,31,33,34)(29,36,37,32)(35,38,39,40);(9,13,14,16)(10,11,15,19)(12,21,22,23)(17,18,20,24)(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=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4841415);