#790812

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

2300 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)(3,8)(4,5)(6,7)(9,10,14,12)(11,16,13,15)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,31)(30,32);(2,7)(5,8)(9,11)(10,15)(12,16)(13,14)(18,23)(19,21)(25,27,29,32)(26,28,31,30);(1,3,6,4)(2,8,7,5)(9,10,14,12)(11,16,13,15)(17,19,22,21)(18,24,23,20)(25,27)(26,30)(28,31)(29,32);(1,4,6,3)(2,8,7,5)(9,12,14,10)(11,16,13,15)(17,20)(18,19)(21,23)(22,24)(25,28,29,30)(26,27,31,32);(1,5,6,8)(2,4,7,3)(9,13,14,11)(10,15,12,16)(17,21,22,19)(18,24,23,20)

Polycyclic presentation:

2^9|a^2=gh,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, 698008);