#1368949

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

1570 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fg,d^2=gh,e^2=hi,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, 1276145);