#4467563

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
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

2193 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)(5,6,9,7)(8,10,12,11)(14,15)(16,20)(21,22,27,25)(23,30,28,24)(26,33,35,29)(31,32,34,36);(6,7)(8,12)(14,15)(18,19)(21,23,27,28)(22,24,25,30)(26,34,35,31)(29,36,33,32);(6,7)(10,11)(13,14,17,15)(16,18,20,19)(21,24,27,30)(22,28,25,23)(26,32,35,36)(29,34,33,31);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18)(21,25,27,22)(23,24,28,30)(26,33,35,29)(31,32,34,36);(5,8)(6,10)(7,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,26)(22,29)(23,31)(24,32)(25,33)(27,35)(28,34)(30,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=h,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 4374759);