#581539

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1766 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fh,d^2=fhi,e^2=fgh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 488735);