#110741

order	512
nilpotency class	3
isoclinism	isoclinic class of #9331
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

578 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17937);