#5022359

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
automorphism	Aut(G) order 2097152
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1616 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)(3,8)(4,6)(5,7)(9,10)(11,12)(13,15)(14,16)(17,18,20,22)(19,21,23,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,37)(38,39);(9,11,14,13)(10,12,16,15)(19,23)(21,24)(26,31)(27,32)(33,39)(35,37);(1,3,6,5)(2,7,4,8)(9,11,14,13)(10,15,16,12)(17,19)(18,21)(20,23)(22,24)(25,27)(26,28)(29,37)(30,32)(31,34)(33,40)(35,38)(36,39);(1,4,6,2)(3,7,5,8)(9,12,14,15)(10,11,16,13)(25,28)(26,32)(27,31)(29,36)(30,34)(33,37)(35,39)(38,40);(1,5,6,3)(2,7,4,8)(9,13,14,11)(10,15,16,12)(17,18)(19,21)(20,22)(23,24)(25,29,30,38)(26,33,31,39)(27,35,32,37)(28,36,34,40)

Polycyclic presentation:

2^9|a^2=fh,b^2=fgi,c^2=fg,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,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, 4929555);