#5560946

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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

1709 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)(9,10)(11,13)(12,14)(15,16)(18,20)(22,23)(25,26,31,28)(27,34,29,32)(30,33,38,36)(35,40,37,39);(12,16)(14,15)(18,20)(22,23)(25,27,31,29)(26,32,28,34)(30,37,38,35)(33,40,36,39);(10,11)(14,15)(17,18,21,20)(19,23,24,22)(25,28,31,26)(27,32,29,34)(30,33,38,36)(35,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19,21,24)(18,22,20,23)(25,29,31,27)(26,32,28,34)(30,37,38,35)(33,39,36,40);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,23,24,22)(25,30,31,38)(26,33,28,36)(27,35,29,37)(32,39,34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,d^2=fg,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5468142);