#108608

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

444 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 162 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)(5,6,9,15,11,16,10,14)(7,17,8,12,19,13,18,20)(21,22,25,31,27,30,26,32)(23,33,36,43,37,28,35,44)(24,38,40,29,41,46,39,45)(34,48,49,42,51,52,50,47);(5,7,10,18,11,19,9,8)(6,12,15,13,16,20,14,17)(21,23,26,36,27,37,25,35)(22,28,32,44,30,33,31,43)(24,34,40,50,41,51,39,49)(29,42,46,48,45,47,38,52);(2,4)(5,8,9,19,11,18,10,7)(6,13,14,12,16,17,15,20)(21,24)(22,29,30,45)(23,34)(25,39)(26,40)(27,41)(28,42,33,47)(31,38,32,46)(35,49)(36,50)(37,51)(43,52,44,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,f^b=fhi,f^c=fhi

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, 15804);