#4318832

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2181 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=f,d^2=fgh,e^2=gh,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, 4226028);