#4057267

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1740 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)(3,7,8,5)(11,16)(13,15)(17,18,20,22)(19,23,24,21)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,38)(39,40);(3,8)(5,7)(9,10,14,12)(11,13,16,15)(17,18,20,22)(19,21,24,23)(26,31)(28,35)(33,40)(36,38);(1,2)(3,5)(4,6)(7,8)(9,11,14,16)(10,15,12,13)(17,19)(18,21)(20,24)(22,23)(25,27,30,32)(26,28,31,35)(29,37,39,34)(33,38,40,36);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,13,16,15)(25,28)(26,32)(27,31)(29,38)(30,35)(33,34)(36,39)(37,40);(1,3)(2,5)(4,8)(6,7)(9,13)(10,11)(12,16)(14,15)(17,18)(19,21)(20,22)(23,24)(25,29,30,39)(26,33,31,40)(27,34,32,37)(28,36,35,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=fgi,c^2=fh,d^2=h,e^2=f,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, 3964463);