#2498482

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1654 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)(9,10,14,16)(11,12,13,15)(18,19)(20,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(9,11,14,13)(10,15,16,12)(17,18,21,19)(20,22,24,23)(26,30)(27,33)(32,35)(34,39);(10,16)(11,13)(17,18,21,19)(20,23,24,22)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,19,21,18)(20,23,24,22)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(1,3,4,7)(2,5,6,8)(9,13,14,11)(10,12,16,15)(17,20)(18,22)(19,23)(21,24)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40)

Polycyclic presentation:

2^9|a^2=fhi,b^2=g,c^2=fh,d^2=gh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2405678);