#5209426

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1431 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fhi,c^2=gi,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 5116622);