#561583

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

2608 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)(10,14)(12,15)(17,18)(19,25)(20,22)(21,28)(23,26)(24,29)(27,30)(31,32);(1,2,3,4)(5,7,8,6)(10,14)(11,16)(18,20)(19,26)(21,29)(30,31);(1,2)(3,4)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(9,11,13,16)(10,15,14,12)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31);(9,12,13,15)(10,16,14,11)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=g,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, 468779);