#5035192

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=gi,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef

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, 4942388);