#2804681

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
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

1698 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=fgh,d^2=fh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2711877);