#4133754

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=fg,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

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, 4040950);