#2568622

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

1617 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)(25,26,30,31)(27,29,33,38)(28,35,36,32)(34,40,39,37);(9,10,13,11)(12,14,16,15)(18,21)(22,24)(26,31)(27,33)(28,36)(37,40);(9,10,13,11)(12,15,16,14)(17,18)(19,22)(20,21)(23,24)(25,27)(26,29)(28,34)(30,33)(31,38)(32,37)(35,40)(36,39);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(25,28,30,36)(26,32,31,35)(27,34,33,39)(29,37,38,40);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,19,20,23)(18,22,21,24)(25,29)(26,33)(27,31)(28,37)(30,38)(32,39)(34,35)(36,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=g,d^2=i,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, 2475818);