#309603

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#10228	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

680 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=h,d^2=fhi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,e^a=eh,c^b=cg,d^b=di,e^b=eh,g^b=gi,g^c=gi,e^d=eh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 216799);