#5394358

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1685 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgh,c^2=fg,d^2=g,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,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, 5301554);