#790813

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2114 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)(3,7,4,8)(9,10)(11,13)(12,16)(14,15)(18,21)(19,23)(25,26,28,30)(27,29,31,32);(2,5)(7,8)(9,11,14,16)(10,12,15,13)(18,21)(22,24);(1,3,6,4)(2,7,5,8)(9,12,14,13)(10,11,15,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,29)(28,30)(31,32);(1,4,6,3)(2,7,5,8)(9,10)(11,12)(13,16)(14,15)(17,19,20,23)(18,22,21,24)(25,27,28,31)(26,29,30,32);(1,5,6,2)(3,8,4,7)(9,13,14,12)(10,11,15,16)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fhi,c^2=f,d^2=fg,e^2=f,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, 698009);