#1324516

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

1720 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=i,d^2=ghi,e^2=g,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, 1231712);