#109429

order	512
nilpotency class	3
isoclinism	isoclinic class of #9755
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 16625);