#109080

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

544 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 230 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eh,d^2=hi,e^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16276);