#3184352

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=ghi,d^2=ghi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 3091548);