#829493

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2181 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 736689);