#4379618

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1640 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=hi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4286814);