#2024176

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
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

1601 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)(3,5,8,7)(10,12)(15,16)(19,23)(22,24)(25,26,29,32)(27,33,34,30)(28,36,37,31)(35,38,40,39);(1,2)(3,7)(4,6)(5,8)(9,10,14,12)(11,15,13,16)(18,21)(22,24)(27,34)(30,33)(35,40)(38,39);(1,2,4,6)(3,5,8,7)(9,11,14,13)(10,15,12,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,31)(29,32)(33,34)(35,38)(36,37)(39,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,15,13,16)(17,19,20,23)(18,22,21,24)(25,27,29,34)(26,30,32,33)(28,35,37,40)(31,38,36,39);(1,3,4,8)(2,5,6,7)(9,13,14,11)(10,15,12,16)(25,28,29,37)(26,31,32,36)(27,35,34,40)(30,38,33,39)

Polycyclic presentation:

2^9|a^2=fhi,b^2=ghi,c^2=f,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 1931372);