#554813

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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

1834 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)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,30)(24,27)(25,33)(26,29)(28,34)(31,35)(32,36);(6,9)(7,11)(14,18)(16,19)(21,23,26,24)(22,27,29,30)(25,31,34,32)(28,35,33,36);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24,26,23)(22,27,29,30)(25,32,34,31)(28,35,33,36);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,19)(18,20)(21,25,26,34)(22,28,29,33)(23,31,24,32)(27,35,30,36);(5,7,8,11)(6,10,9,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,35)(33,34)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=fghi,e^2=h,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,e^b=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, 462009);