#763104

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1830 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=hi,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, 670300);