#540369

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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

2258 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 447565);