#560422

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

2366 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=g,d^2=f,e^2=f,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, 467618);