#1605689

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2449 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40);(6,8)(10,11)(13,21)(15,17)(19,24)(26,30)(27,34)(28,37)(38,39);(5,6,7,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40);(1,2)(3,4)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=fg,c^2=gh,d^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, 1512885);