#107235

order	512
nilpotency class	3
isoclinism	isoclinic class of #107214
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

695 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=di,e^b=ehi,f^b=fi,d^c=dhi,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14431);