#442177

order	512
nilpotency class	3
isoclinism	isoclinic class of #442176
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15702
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

915 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,12,8,13,6,28)(3,14,17,9,18,35,29,34)(4,19,22,38,23,10,31,39)(5,24,26,41,27,11,30,42)(15,44,20,32,47,36,46,57)(16,40,48,58,49,43,25,33)(21,37,54,60,55,50,53,61)(45,59,51,64,52,62,63,56);(1,3,7,17,8,18,6,29)(2,9,12,35,13,34,28,14)(4,20,22,47,23,46,31,15)(5,25,26,16,27,48,30,49)(10,36,39,57,19,44,38,32)(11,40,42,58,24,43,41,33)(21,52,54,63,55,45,53,51)(37,56,60,59,50,64,61,62);(1,4,8,23)(2,10,13,19)(3,15,18,47)(5,21,27,55)(6,22,7,31)(9,32,34,57)(11,37,24,50)(12,38,28,39)(14,36,35,44)(16,45,49,52)(17,46,29,20)(25,51,48,63)(26,53,30,54)(33,56,58,64)(40,62,43,59)(41,60,42,61);(1,5)(2,11,13,24)(3,16)(4,21)(6,30)(7,26)(8,27)(9,33,34,58)(10,37,19,50)(12,42,28,41)(14,43,35,40)(15,45)(17,48)(18,49)(20,51)(22,54)(23,55)(25,29)(31,53)(32,56,57,64)(36,59,44,62)(38,61,39,60)(46,63)(47,52)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^c=gi,e^d=ei,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 349373);