#763207

order	512
nilpotency class	2
isoclinism	isoclinic class of #759288
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

2292 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,c^2=h,d^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,e^c=ef

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, 670403);