#434020

order	512
nilpotency class	3
isoclinism	isoclinic class of #434019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15495
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

907 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,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, 341216);