#587478

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

2354 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)(6,7)(8,12)(14,16)(18,20)(21,22)(23,29)(24,26)(25,32)(27,30)(28,33)(31,34)(35,36);(1,3,4,2)(6,7)(10,11)(15,19)(18,20)(22,24)(23,30)(25,33)(34,35);(5,6,9,7)(8,10,12,11)(13,14,17,16)(15,18,19,20)(21,23)(22,27)(24,29)(25,31)(26,30)(28,34)(32,35)(33,36);(5,7,9,6)(8,11,12,10)(13,15,17,19)(14,18,16,20)(21,24,26,22)(23,27,30,29)(25,32,33,28)(31,34,36,35);(5,8,9,12)(6,10,7,11)(13,16,17,14)(15,20,19,18)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=g,e^2=fg,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, 494674);