#600851

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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

1978 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)(7,11,12,9)(13,14,18,16)(15,19,17,20)(21,22,26,27)(23,25,29,34)(24,28,33,35)(30,32,31,36);(5,6,8,10)(7,9,12,11)(13,15,18,17)(14,19,16,20)(22,27)(25,34)(28,35)(32,36);(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,15,18,17)(14,20,16,19)(21,23,26,29)(22,25,27,34)(24,31,33,30)(28,36,35,32);(5,6)(7,9)(8,10)(11,12)(13,16,18,14)(15,19,17,20)(21,24,26,33)(22,28,27,35)(23,30,29,31)(25,32,34,36);(1,3,4,2)(5,7,8,12)(6,9,10,11)(13,17,18,15)(14,19,16,20)(21,25)(22,29)(23,27)(24,32)(26,34)(28,31)(30,35)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=fgh,e^2=fi,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 508047);