#1613998

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2214 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=h,d^2=fgi,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, 1521194);