#110689

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

606 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17885);