#5482164

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1694 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)(19,23)(22,24)(25,26,30,29)(27,31,33,37)(28,35,36,32)(34,40,39,38);(3,7)(6,8)(9,11,12,15)(10,13,14,16)(18,21)(22,24)(26,29)(27,33)(32,35)(34,39);(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,31)(28,34)(29,37)(30,33)(32,38)(35,40)(36,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(25,28,30,36)(26,32,29,35)(27,34,33,39)(31,38,37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,22)(20,23)(21,24)(25,29,30,26)(27,31,33,37)(28,35,36,32)(34,38,39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=g,d^2=f,e^2=f,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, 5389360);