#280096

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

706 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)(3,5)(4,6)(7,8)(17,18)(19,25)(20,28)(21,23)(22,26)(24,27)(29,32)(30,31);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(14,16)(18,23)(19,21)(24,28)(25,26)(27,30)(31,32);(2,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,19,22,21)(18,23,26,25)(20,27,29,30)(24,28,31,32);(10,14)(13,16)(17,20,22,29)(18,24,26,31)(19,27,21,30)(23,28,25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 187292);