#1572835

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

2162 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)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26,30,28)(27,33,35,31)(29,36,38,32)(34,39,37,40);(6,8)(10,14)(12,20)(15,17)(18,24)(25,27,30,35)(26,31,28,33)(29,37,38,34)(32,40,36,39);(5,6,7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,40)(37,38);(1,2)(3,4)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,30,26)(27,31,35,33)(29,36,38,32)(34,39,37,40);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,d^2=f,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, 1480031);