#109069

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#382
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

492 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

(1,2)(3,4)(5,6,8,13,10,14,9,12)(7,15,16,11,18,19,17,20)(21,22)(23,32,36,46)(24,37)(25,30)(26,41)(27,31)(28,34,43,35)(29,39)(33,47,50,42)(38,44)(40,51)(45,48,52,49);(2,4)(5,7,9,17,10,18,8,16)(6,11,13,19,14,20,12,15)(21,23)(22,28,31,43)(24,33)(25,34)(26,35)(27,36)(29,42,44,47)(30,32,41,46)(37,45,51,52)(38,48)(39,49)(40,50);(21,24,26,39,27,40,25,38)(22,29,30,37,31,44,41,51)(23,33,35,49,36,50,34,48)(28,42,32,45,43,47,46,52)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=hi,e^2=i,f^2=h,g^2=h,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, 16265);