#116372

order	512
nilpotency class	3
isoclinism	isoclinic class of #116371
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,e^2=i,f^2=h,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=ei,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23568);