#1323726

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

1498 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=g,c^2=fi,d^2=gi,e^2=fg,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, 1230922);