#712926

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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fhi,c^2=fi,d^2=fg,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, 620122);