#5491400

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 1048576
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

2388 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,16,14,12)(17,18,20,22)(19,21,23,24)(25,26,30,33)(27,31,28,35)(29,32,37,40)(34,38,36,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,14)(15,16)(19,23)(21,24)(25,27,30,28)(26,31,33,35)(29,36,37,34)(32,39,40,38);(2,6)(3,8)(9,12,13,16)(10,14,15,11)(17,19)(18,21)(20,23)(22,24)(25,28,30,27)(26,31,33,35)(29,36,37,34)(32,38,40,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,16)(14,15)(25,26)(27,31)(28,35)(29,32)(30,33)(34,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,29,30,37)(26,32,33,40)(27,34,28,36)(31,38,35,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=fi,e^2=fg,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, 5398596);