#535999

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 443195);