#3856600

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1706 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)(3,7)(9,10,13,12)(11,15,16,14)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,38)(37,39);(2,5)(6,8)(17,18,21,20)(19,23,24,22)(26,30)(27,34)(32,36)(35,37);(1,2)(3,6)(4,5)(7,8)(9,10,13,12)(11,14,16,15)(17,19,21,24)(18,22,20,23)(25,26)(27,31)(28,36)(29,30)(32,38)(33,34)(35,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,12,15)(17,18,21,20)(19,22,24,23)(25,27,29,34)(26,31,30,33)(28,37,38,35)(32,40,36,39);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,20,21,18)(19,23,24,22)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,40)(34,37)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fgh,c^2=hi,d^2=fhi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,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, 3763796);