#535368

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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

2366 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fg,e^2=g,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, 442564);