#737654

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1910 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)(3,8)(4,7)(5,6)(9,10)(11,12)(13,16)(14,15)(17,18,21,20)(19,22,24,23)(27,32)(28,31);(2,6)(4,8)(9,11,13,15)(10,14,16,12)(18,20)(22,23)(25,26,30,29)(27,31,32,28);(2,6)(3,7)(9,12,13,14)(10,15,16,11)(17,18,21,20)(19,23,24,22)(25,27,30,32)(26,31,29,28);(1,3,5,7)(2,4,6,8)(9,12)(10,11)(13,14)(15,16)(17,19,21,24)(18,22,20,23)(25,28)(26,27)(29,32)(30,31);(1,4)(2,7)(3,6)(5,8)(9,11)(10,12)(13,15)(14,16)(17,20,21,18)(19,23,24,22)(25,29,30,26)(27,28,32,31)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=fi,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,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, 644850);