#580604

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

1942 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)(7,9)(8,11)(10,12)(21,22)(23,29)(24,31)(25,33)(26,27)(28,35)(30,32)(34,36);(1,2,4,3)(5,7,10,11)(6,8,12,9)(14,17)(18,20)(22,27)(24,29)(28,33)(32,36);(5,8,10,9)(6,11,12,7)(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,34)(26,31)(27,29)(28,36)(30,35)(32,33);(1,3,4,2)(5,9,10,8)(6,11,12,7)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,34)(35,36);(5,6)(7,8)(9,11)(10,12)(13,15,16,19)(14,18,17,20)(21,25,26,35)(22,28,27,33)(23,30,31,34)(24,32,29,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fg,d^2=fgi,e^2=fh,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, 487800);