#1310256

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1428 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fhi,c^2=fgh,d^2=fgi,e^2=fh,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, 1217452);