#5069163

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1663 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)(4,8)(5,6)(10,13)(14,16)(17,18,20,21)(19,23,24,22)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,37)(38,39);(1,3,6,8)(2,4,5,7)(11,15)(14,16)(18,21)(22,23)(26,28)(29,37)(31,33)(34,38);(2,5)(3,8)(9,10,12,13)(11,14,15,16)(17,18,20,21)(19,23,24,22)(25,27)(26,31)(28,33)(29,38)(30,35)(32,40)(34,37)(36,39);(1,4)(2,3)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,40,38,39);(1,5,6,2)(3,4,8,7)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,24)(21,23)(25,29,30,37)(26,32,28,36)(27,34,35,38)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=fg,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,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, 4976359);