#576187

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

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

2062 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 483383);