#586272

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1900 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)(5,6,9,7)(8,11,12,10)(13,14,16,17)(15,19,20,18)(22,24)(27,29)(28,32)(34,35);(6,7)(8,12)(14,17)(15,20)(21,22,26,24)(23,29,31,27)(25,32,33,28)(30,34,36,35);(6,7)(10,11)(13,15,16,20)(14,18,17,19)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,20)(17,19)(21,24,26,22)(23,27,31,29)(25,32,33,28)(30,34,36,35);(5,8)(6,10)(7,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,25,26,33)(22,28,24,32)(23,30,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=h,d^2=gh,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, 493468);