#2548837

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1681 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=fgh,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, 2456033);