#1274445

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1710 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=f,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1181641);