#109295

order	512
nilpotency class	3
isoclinism	isoclinic class of #109288
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₄⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

510 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)(3,9)(4,7)(5,13)(6,10)(8,12)(11,14)(15,16)(17,18)(19,27,30,23)(20,31)(21,35)(22,25)(24,33)(26,32)(28,41,44,45)(29,37,42,40)(34,46)(36,43,39,48)(38,47);(1,3,5,11,4,10,12,16)(2,6,8,14,7,9,13,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,27)(26,37)(31,45)(32,42)(33,43)(34,44)(35,40)(38,39)(41,46)(47,48);(2,7)(6,9)(8,13)(14,15)(17,20,32,47,22,34,21,33)(18,24,35,46,25,38,26,31)(19,28,42,48,30,44,29,43)(23,36,40,41,27,39,37,45)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ehi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,f^b=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, 16491);