#556221

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

2078 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fgi,d^2=h,e^2=f,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, 463417);