#5022376

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1926 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:

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

Polycyclic presentation:

2^9|a^2=fgh,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, 4929572);