#563094

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

2652 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)(10,13)(14,16)(17,18,23,20)(19,25,21,24)(22,26,30,28)(27,32,29,31);(6,8)(11,15)(14,16)(17,19,23,21)(18,24,20,25)(22,29,30,27)(26,32,28,31);(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31);(9,10)(11,14)(12,13)(15,16)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=gi,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, 470290);