#581519

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

1770 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)(5,6,8,10)(7,11,12,9)(13,14,17,16)(15,18,20,19)(21,22,26,27)(23,24,29,32)(25,28,34,35)(30,31,33,36);(1,2,4,3)(5,7,8,12)(6,9,10,11)(13,14,17,16)(15,19,20,18)(22,27)(23,29)(28,35)(30,33);(6,10)(9,11)(13,15,17,20)(14,18,16,19)(21,23)(22,24)(25,33)(26,29)(27,32)(28,36)(30,34)(31,35);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(21,24)(22,23)(25,31)(26,32)(27,29)(28,30)(33,35)(34,36);(5,6)(7,9)(8,10)(11,12)(13,16,17,14)(15,19,20,18)(21,25,26,34)(22,28,27,35)(23,30,29,33)(24,31,32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fgh,d^2=ghi,e^2=h,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, 488715);