#4886763

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

1918 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)(11,15)(13,16)(17,18,20,22)(19,21,23,24)(25,26)(27,33)(28,36)(29,34)(30,31)(32,37)(35,39)(38,40);(1,2,4,6)(3,5,8,7)(9,10,12,14)(11,13,15,16)(19,23)(21,24)(26,31)(27,34)(28,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,20,23)(18,21,22,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(17,18)(19,21)(20,22)(23,24)(25,28)(26,32)(27,35)(29,39)(30,37)(31,36)(33,38)(34,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,29)(26,27)(28,38)(30,33)(31,34)(32,40)(35,36)(37,39)

Polycyclic presentation:

2^9|a^2=fi,b^2=fghi,c^2=gi,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, 4793959);