#5490527

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1364 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5397723);