#105777

order	512
nilpotency class	3
isoclinism	isoclinic class of #8465
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,6,10)(3,7,9,5)(4,8)(11,14)(12,15)(13,16)(17,18,21,27,23,26,22,28)(19,29,31,41,32,24,36,39)(20,33,34,42,35,25,37,40)(30,38,44,47,46,43,45,48);(1,3,7,2,6,9,5,10)(4,11,12,8,13,15,14,16)(17,19,22,31,23,32,21,36)(18,24,28,39,26,29,27,41)(20,30,34,45,35,46,37,44)(25,38,40,48,33,43,42,47);(1,4)(2,8)(3,11)(5,14)(6,13)(7,12)(9,15)(10,16)(17,20)(18,25)(19,30)(21,37)(22,34)(23,35)(24,38)(26,33)(27,42)(28,40)(29,43)(31,45)(32,46)(36,44)(39,48)(41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12973);