#606386

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

1914 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)(5,6,9,7)(8,11,12,10)(13,14,18,16)(15,19,17,20)(22,26)(23,29)(24,31)(34,36);(6,7)(8,12)(13,15,18,17)(14,19,16,20)(22,26)(23,29)(28,33)(30,35);(1,2,4,3)(6,7)(10,11)(13,16,18,14)(15,19,17,20)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36);(1,2,4,3)(5,7,9,6)(8,11,12,10)(13,17,18,15)(14,19,16,20)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(1,3,4,2)(5,8,9,12)(6,10,7,11)(21,24)(22,28)(23,30)(25,31)(26,33)(27,34)(29,35)(32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fgh,c^2=fi,d^2=fgi,e^2=fghi,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, 513582);