#592005

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

1788 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)(5,6,9,8)(7,10,11,12)(14,15)(16,20)(23,29)(26,32)(30,35)(33,36);(6,8)(10,12)(14,15)(18,19)(21,22,25,28)(23,26,29,32)(24,27,31,34)(30,33,35,36);(1,3,4,2)(7,11)(10,12)(13,14,17,15)(16,18,20,19)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,34)(35,36);(5,7,9,11)(6,10,8,12)(13,15,17,14)(16,19,20,18)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(5,8,9,6)(7,12,11,10)(13,16)(14,18)(15,19)(17,20)(21,24,25,31)(22,27,28,34)(23,30,29,35)(26,33,32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fgi,d^2=fg,e^2=fgh,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, 499201);