#109324

order	512
nilpotency class	3
isoclinism	isoclinic class of #109288
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

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×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

475 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,e^2=h,f^2=hi,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, 16520);