#4912362

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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

1524 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)(10,15)(11,12)(17,18,20,22)(19,23,24,21)(25,26)(27,33)(28,36)(29,32)(30,35)(31,38)(34,39)(37,40);(9,10)(11,16)(12,13)(14,15)(19,24)(21,23)(25,26,29,32)(27,30,35,33)(28,31,38,36)(34,39,37,40);(9,11,14,12)(10,16,15,13)(17,18)(19,21)(20,22)(23,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,16,15,13)(17,19,20,24)(18,21,22,23)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(1,3,4,7)(2,6,5,8)(9,13,14,16)(10,12,15,11)(25,28,29,38)(26,31,32,36)(27,34,35,37)(30,39,33,40)

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=fh,d^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4819558);