#4084465

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
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

1606 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fhi,d^2=h,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=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, 3991661);