#646818

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2388 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 554014);