#557389

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fhi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=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, 464585);