#2568636

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
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

1544 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fg,d^2=fi,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2475832);