#516423

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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

2846 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)(10,14)(15,16)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,31,32,30);(2,4)(10,14)(15,16)(18,19)(23,25)(24,26)(30,31);(1,2)(3,4)(9,10)(11,15)(12,16)(13,14)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31);(9,11,13,12)(10,15,14,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(9,12,13,11)(10,16,14,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 423619);