#4121498

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1526 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4028694);