#3163859

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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

1860 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fh,d^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3071055);