#582318

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1866 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,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, 489514);