#606113

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

2756 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)(6,9)(10,11)(13,14,18,20)(15,17,16,19)(21,22,25,28)(23,26,30,33)(24,27,31,34)(29,32,35,36);(5,6)(7,11)(8,9)(10,12)(13,15)(14,19)(16,18)(17,20)(21,23,25,30)(22,26,28,33)(24,29,31,35)(27,32,34,36);(1,2,4,3)(6,9)(10,11)(14,20)(15,16)(21,23)(22,26)(24,29)(25,30)(27,32)(28,33)(31,35)(34,36);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,17,20,19)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,34)(35,36);(1,3,4,2)(5,7)(6,10)(8,12)(9,11)(13,17)(14,15)(16,20)(18,19)(21,24,25,31)(22,27,28,34)(23,29,30,35)(26,32,33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fghi,d^2=fhi,e^2=gi,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, 513309);