#580660

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2178 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fghi,c^2=f,d^2=i,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, 487856);