#582671

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

1964 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)(15,19)(18,20)(21,22,25,26)(23,29,31,27)(24,32,33,28)(30,34,36,35);(5,6,9,7)(8,10,12,11)(14,17)(18,20)(22,26)(27,29)(28,32)(34,35);(1,3,4,2)(6,7)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23,25,31)(22,27,26,29)(24,30,33,36)(28,34,32,35);(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,19)(17,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(5,8)(6,10)(7,11)(9,12)(21,24,25,33)(22,28,26,32)(23,30,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=fg,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, 489867);