#2568638

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1677 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=g,d^2=i,e^2=fhi,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, 2475834);