#2571384

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

1810 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=gh,d^2=fgi,e^2=ghi,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, 2478580);