#433544

order	512
nilpotency class	3
isoclinism	isoclinic class of #427666
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2527
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15484
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

747 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)(3,8,15,31)(4,17)(5,21,23,40)(6,12)(7,11)(9,19)(10,24,36,27)(13,29,42,33)(14,44)(16,32,25,38)(18,49,51,60)(20,39)(22,54)(26,35)(28,37)(30,47)(34,53,48,52)(41,61)(43,56,46,58)(45,57)(50,64)(55,63)(59,62);(1,3)(2,8,11,31)(4,13,19,42)(5,22,23,14)(6,25)(7,15)(9,33)(10,30)(12,38,37,32)(16,28)(17,29)(18,50)(20,46,26,43)(21,54)(24,45,27,47)(34,59,48,55)(35,56)(36,57)(39,58)(40,44)(41,51)(49,64,60,61)(52,62)(53,63);(1,4)(2,9,11,17)(3,13,15,42)(5,18)(6,26)(7,19)(8,29)(10,34,36,48)(12,39,37,35)(14,41,22,50)(16,46,25,43)(20,28)(21,49,40,60)(23,51)(24,53)(27,52)(30,55)(31,33)(32,58)(38,56)(44,64)(45,63,47,62)(54,61)(57,59);(1,5)(2,10)(3,14)(4,18)(6,27)(7,23)(8,30)(9,34)(11,36)(12,40)(13,41)(15,22)(16,47)(17,48)(19,51)(20,53)(21,37)(24,28)(25,45)(26,52)(29,55)(31,57)(32,54)(33,59)(35,49)(38,44)(39,60)(42,50)(43,63)(46,62)(56,64)(58,61)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340740);