#5479933

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 52 points:

(2,4)(6,10)(7,13)(12,17)(15,19)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34)(37,38,42,45)(39,46,43,40)(41,44,50,52)(47,51,49,48);(8,16)(12,17)(15,19)(18,20)(23,29)(26,33)(30,35)(34,36)(37,39,42,43)(38,40,45,46)(41,47,50,49)(44,48,52,51);(5,6,9,10)(7,14,13,11)(8,12,16,17)(15,20,19,18)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35)(37,39)(38,40)(41,49)(42,43)(44,51)(45,46)(47,50)(48,52);(1,2)(3,4)(5,7,9,13)(6,11,10,14)(8,15,16,19)(12,18,17,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36)(37,40)(38,43)(39,45)(41,48)(42,46)(44,49)(47,52)(50,51);(5,8,9,16)(6,12,10,17)(7,15,13,19)(11,18,14,20)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)(37,41,42,50)(38,44,45,52)(39,47,43,49)(40,48,46,51)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fg,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, 5387129);