#562927

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

2448 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fg,d^2=i,e^2=g,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, 470123);