#576686

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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

2554 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)(10,13)(14,16)(17,18,21,22)(19,25,27,23)(20,28,30,24)(26,31,29,32);(1,2,3,4)(11,15)(14,16)(18,22)(23,25)(24,28)(31,32);(1,2)(3,4)(9,10)(11,14)(12,13)(15,16)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31);(9,11,12,15)(10,14,13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32);(9,10)(11,14)(12,13)(15,16)(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef

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, 483882);