#515853

order	512
nilpotency class	2
isoclinism	isoclinic class of #35346
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

2704 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)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31);(6,8)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(24,27)(26,31);(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32);(1,3,4,2)(9,11,12,15)(10,13,14,16)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32);(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fgh,d^2=fi,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=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, 423049);