#1287252

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

1862 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=gh,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, 1194448);