#550131

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

2406 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)(11,15)(12,13)(14,16)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30);(1,2,4,3)(6,8)(10,13)(14,15)(20,28)(24,29)(26,32)(30,31);(1,2,4,3)(5,6,7,8)(10,13)(14,15)(17,19,22,27)(18,23,21,25)(20,26,28,32)(24,30,29,31);(9,10)(11,14)(12,13)(15,16)(17,20,22,28)(18,24,21,29)(19,26,27,32)(23,30,25,31);(1,3,4,2)(9,11)(10,14)(12,16)(13,15)(17,21,22,18)(19,25,27,23)(20,29,28,24)(26,31,32,30)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=hi,d^2=fg,e^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 457327);