#5385169

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1456 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)(9,10,15,16)(11,12,14,13)(19,20)(23,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(9,11)(10,13)(12,16)(14,15)(18,22)(23,24)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(9,12,15,13)(10,11,16,14)(17,18)(19,23)(20,24)(21,22)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,13,15,12)(10,11,16,14)(17,19,21,20)(18,23,22,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,3,4,7)(2,5,6,8)(9,14,15,11)(10,12,16,13)(17,20,21,19)(18,24,22,23)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,c^2=g,d^2=gh,e^2=fi,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, 5292365);