#1735940

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1633 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=fi,d^2=fh,e^2=hi,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, 1643136);