#580053

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

2268 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)(6,9)(10,12)(14,18)(16,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(1,2,4,3)(7,11)(10,12)(14,18)(15,19)(22,26)(27,29)(28,32)(34,35);(5,6)(7,10)(8,9)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,27)(24,32)(25,26)(28,33)(29,31)(30,35)(34,36);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,35)(33,36);(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,20)(18,19)(21,24,25,33)(22,28,26,32)(23,30,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fghi,e^2=f,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, 487249);